the booklet of speakers and abstracts - ica 2016ica.balikesir.edu.tr/dosyalar/2013/abstracts.pdf ·...

The International Conference on

Algebra in Honour of Patrick SMITH

and John CLARK’s 70th Birthdays

The Booklet of

Speakers and Abstracts

12-15 AUGUST 2013,

Burhaniye - Balikesir - TURKEY

1

The International Conference on Algebra in Honour of Patrick SMITH and John CLARK’s 70th Birthdays, 12-15 AUGUST, 2013

Honorary Committee• Prof. Dr. Mahir ALKAN (Rector of Balıkesir University)

• Prof. Dr. Ali DOGRAMACI (Chair, Board of Trustees, Bilkent University)

• Prof. Dr. A. Murat TUNCER (Rector of Hacettepe University)

Scientific Committee• Kenneth A. BROWN, University of Glasgow, Glasgow, UK

• Leonid BOKUT, Novosibirsk State University, Novosibirsk, Russia

• Ismail Naci CANGUL, Uludag University, Bursa, Turkey

• A. Sinan CEVIK, Selcuk University, Konya, Turkey

• Yuqun CHEN, South China Normal University, Guangzhou, China

• Yong Uk CHO, Silla University, Pusan, Korea

• Pedro Antonio GUIL ASENSIO, University of Murcia, Murcia, Spain

• Abdullah HARMANCI, Hacettepe University, Ankara, Turkey

• Dolors HERBERA, Autonomous University of Barcelona, Barcelona, Spain

• Derya KESKIN TUTUNCU, Hacettepe University, Ankara, Turkey

• Christian LOMP, University of Porto, Porto, Portugal

• Barbara L. OSOFSKY, Rutgers University, New Brunswick,USA

• A. Cigdem OZCAN, Hacettepe University, Ankara, Turkey

• Recep SAHIN, Balikesir University, Balikesir, Turkey

• Adnan TERCAN, Hacettepe University, Ankara, Turkey

Balikesir University & Hacettepe University 3 Burhaniye - Balikesir - TURKEY


Organizing CommitteeOrganizing Committee Chair;

Adnan TERCAN, Hacettepe University, Ankara, Turkey

• Firat ATES, Balikesir University, Balikesir, Turkey

• Ahmet Sinan CEVIK, Selcuk University, Konya, Turkey

• Sebahattin IKIKARDES, Balikesir University, Balikesir, Turkey

• Derya KESKIN TUTUNCU, Hacettepe University, Ankara, Turkey

• Christian LOMP, University of Porto, Porto, Portugal

• A. Cigdem OZCAN, Hacettepe University, Ankara, Turkey

Local Organizing Committee• Meltem ALTUN, Hacettepe University, Ankara, Turkey

• Pinar AYDOGDU, Hacettepe University, Ankara, Turkey

• Metin BAGDAT, Gazi University, Ankara, Turkey

• Engin BUYUKASIK, Izmir Institue of Technology, Izmir, Turkey

• Canan CELEP YUCEL, Pamukkale University, Denizli, Turkey

• Ozge CELIK, Balikesir University, Balikesir, Turkey

• Merve COLPAN, Balikesir University, Balikesir, Turkey

• Bilal DEMIR, Balikesir University, Balikesir, Turkey

• Ahmet EMIN, Balikesir University, Balikesir, Turkey

• Eylem GUZEL KARPUZ, K. Mehmetbey University, Karaman, Turkey

• Berke KALEBOGAZ, Hacettepe University, Ankara, Turkey

• Yeliz KARA, Hacettepe University, Ankara, Turkey

• M. Tamer KOSAN, Gebze Institute of Technology, Kocaeli, Turkey

• Yeliz KURTULDU, Balikesir University, Balikesir, Turkey

• Engin MERMUT, Dokuz Eylul University, Izmir, Turkey

• Pinar METE, Balikesir University, Balikesir, Turkey

• Bulent SARAC, Hacettepe University, Ankara, Turkey

• Umit SARP, Balikesir University, Balikesir, Turkey

• Ramazan YASAR, Hacettepe University, Ankara, Turkey

• Filiz YILDIZ, Hacettepe University, Ankara, Turkey

• Nazlı YILDIZ IKIKARDES, Balikesir University, Balikesir, Turkey



Invited Speakers• Gary F. BIRKENMEIER, University of Louisiana, Lafayette, USA

• Leonid BOKUT, Novosibirsk State University, Novosibirsk, Russia

• Kenneth A. BROWN, University of Glasgow, Glasgow, UK

• Yuqun CHEN, South China Normal University, Guangzhou, China

• Ali Bulent EKIN, Ankara University, Ankara, Turkey

• Alberto FACCHINI, University of Padova, Padove, Italy

• Surender K. JAIN, King Abdulaziz University,Jeddah,Saudi Arabia,and OhioUniversity,Athens,USA

• Sergio R. LOPEZ-PERMOUTH, Ohio University, Athens, USA

• W. Keith NICHOLSON, University of Calgary, Calgary, Canada

• Barbara L. OSOFSKY, Rutgers University, New Brunswick,USA

• Stephen J. PRIDE, University of Glasgow, Glasgow, UK

• Edmund PUCZYLOWSKI, University of Warsaw, Warszawa, Poland

• Rick THOMAS, University of Leicester, Leicester, UK

• Robert WISBAUER, Heinrich-Heine-Universitt, Dusseldorf, Germany



Contents

1 Invited Speakers 121.1 Gary F. BIRKENMEIER . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Leonid A. BOKUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Kenneth A. BROWN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Yuqun CHEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Noyan ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Alberto FACCHINI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Surender K. JAIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.8 Sergio R. LOPEZ-PERMOUTH . . . . . . . . . . . . . . . . . . . . . . 191.9 W. Keith NICHOLSON . . . . . . . . . . . . . . . . . . . . . . . . . . 201.10 Barbara L. OSOFSKY . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.11 Edmund R.PUCZY LOWSKI . . . . . . . . . . . . . . . . . . . . . . . 221.12 Richard M. THOMAS . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.13 Robert WISBAUER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Contributed Speakers 282.1 Jawad ABUHLAIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Ahmad AL KHALAF . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Emine ALBAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Shakir ALI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Abdollah ALHEVAZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6 Mustafa ALKAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Abu Zaid ANSARI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8 Orest D. ARTEMOVYCH . . . . . . . . . . . . . . . . . . . . . . . . . 362.9 Umamaheswaran ARUNACHALAM . . . . . . . . . . . . . . . . . . . 382.10 Mohammad ASHRAF . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.11 Yıldız AYDIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.12 Ayman BADAWI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.13 Evgenii BASHKIROV . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.14 Samruam BAUPRADIST . . . . . . . . . . . . . . . . . . . . . . . . . 432.15 Firdhousi BEGUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.16 Vijay Kumar BHAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.17 Ismail Naci CANGUL . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.18 Paula A.A.B. CARVALHO . . . . . . . . . . . . . . . . . . . . . . . . 472.19 Canan Celep YUCEL . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.20 Jianlong CHEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.21 Jules Clement MBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.22 Secil CEKEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.23 Nadeem Ahmad DAR . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.24 Kinkar Ch. DAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.25 Ali Ahmed DAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.26 Fatemeh DEHGHANI ZADEH . . . . . . . . . . . . . . . . . . . . . . 552.27 Cagri DEMIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.28 Nanqing DING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.29 Yilmaz DURGUN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.30 Ozgur EGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.31 Temha ERKOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.32 Sehmus FINDIK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64



2.33 Silvia Claudia GAVITO . . . . . . . . . . . . . . . . . . . . . . . . . . 662.34 Hanieh GOLMAKANI . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.35 Mouloud GOUBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.36 Nico J. GROENEWALD . . . . . . . . . . . . . . . . . . . . . . . . . . 702.37 Eylem GUZEL KARPUZ . . . . . . . . . . . . . . . . . . . . . . . . . 712.38 Serpil GUNGOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.39 Orhan GURGUN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.40 Claus HAETINGER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.41 Khalid Ali Mohammad HAMDIN . . . . . . . . . . . . . . . . . . . . . 762.42 Ayazul HASAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.43 Can HATIPOGLU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.44 Ilker INAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.45 Berke KALEBOGAZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.46 Tolga KARAYAYLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.47 Abidin KAYA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.48 Shahoor KHAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.49 Sibel KILICARSLAN CANSU . . . . . . . . . . . . . . . . . . . . . . 852.50 M. Tamer KOSAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.51 Berna KOSAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.52 Yosum KURTULMAZ . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.53 Semra KUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.54 Christian LOMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.55 Adolf MADER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.56 Phool MIYAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.57 Najat Mohammed MUTHANA . . . . . . . . . . . . . . . . . . . . . . 942.58 Sinem ODABASI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.59 Seda OGUZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.60 Salahattin OZDEMIR . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.61 Gulcan OZKUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.62 Talat PARVEEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.63 Manoj Kumar PATEL . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.64 Hourong QIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.65 Udhayakumar RAMALINGAM . . . . . . . . . . . . . . . . . . . . . . 1032.66 Kulumani M. RANGASWAMY . . . . . . . . . . . . . . . . . . . . . . 1042.67 Nadeem ur REHMAN . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.68 P. G. ROMEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.69 Mohammad SADEK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.70 Ali M. SAGER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.71 Liang SHEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.72 Faiza SHUJAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.73 Ebru SOLAK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.74 Carlos SONEIRA CALVO . . . . . . . . . . . . . . . . . . . . . . . . . 1132.75 David SSEVVIIRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142.76 Serap SAHINKAYA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.77 Yahya TALEBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.78 Bertha TOME-ARREOLA . . . . . . . . . . . . . . . . . . . . . . . . 1182.79 Hamid USEFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.80 Burcu UNGOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.81 Lia VAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212.82 Indah Emilia WIJAYANTI . . . . . . . . . . . . . . . . . . . . . . . . 122



2.83 Feyza YALCIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.84 Ece YETKIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.85 Erol YILMAZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262.86 Utku YILMAZTURK . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282.87 Figen YUZBASI ERYILMAZ . . . . . . . . . . . . . . . . . . . . . . . 1292.88 Parvaneh ZOLFAGHARI . . . . . . . . . . . . . . . . . . . . . . . . . 131

3 List of Participants 134


Invited Speakers


1 Invited Speakers

1.1 Gary F. BIRKENMEIER

Generalizations of the Extending Property

Gary F. BIRKENMEIER

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA70504-1010, USA,

[email protected]

Abstract: Recall that a module M is called ”extending or CS” if every submodule isessential in a direct summand of M. In this talk various generalizations of the conceptof an extending (also called a CS or C1) module are presented. These generalizationsinclude the notions of C11-modules, FI-extending modules, G-extending modules, etc.



1.2 Leonid A. BOKUT

Some Recent Results on Grobner-Shirshov Bases

Leonid A. BOKUT

School of Mathematical Sciences, South China Normal University, Guangzhou510631, P.R. China

Sobolev Institute of Mathematics, Russian Academy of Sciences, Siberian Branch,Novosibirsk 630090, Russia

[email protected]

Abstract: I will speak on Grobner-Shirshov bases theory for Lie algebras over com-mutative algebras, plactic monoids, categories, semirings, strict monoidal categories.These are joint results with Yuqun Chen, Yangshan Chen, Yu Li, Qiuhui Mo, JingLiu, Wendy Chen from the SCNU, Guangzhou.

Keywords: Grobner-Shirshov basis, Lie algebra, category

2010 Mathematics Subject Classification: 13P10; 17B01; 18D10



1.3 Kenneth A. BROWN

Noncommutative Unipotent Groups

Kenneth A. BROWN

School of Mathematics and Statistics, University of Glasgow, [email protected]

Abstract: I will review some recent results of myself and collaborators on noncom-mutative Hopf algebras, connecting them to some classical topics in noncommutativering theory and in algebraic groups. I will explain relevant background in the talk.



1.4 Yuqun CHEN

Grobner-Shirshov Bases for Strict Monoidal Categories

Yuqun CHEN

School of Mathematical Sciences, South China Normal University, Guangzhou510631, P.R. [email protected]

Abstract: We establish Grobner-Shirshov bases theory for strict monoidal categoriesand give some applications.

Keywords: Grobner-Shirshov basis, category

2010 Mathematics Subject Classification: 13P10; 18D10



1.5 Noyan ER

A Kernel Functor Defined Via Relative Injectivity AndApplications

Noyan ER

Department of Mathematics, University of Rio Grande, Rio Grande, OH 45674, [email protected]

Abstract: The recent work on modules with minimal injectivity domains by authorsincluding Alahmadi, Alkan, Er, Lopez-Permouth, and Sokmez is part of the inspirationfor this talk. The functor mentioned in the title will be applied to characterize somerings and to answer some questions inspired by Torsion Theory.



1.6 Alberto FACCHINI

Some Noteworthy Ideals in Categories of Modules

Alberto FACCHINI

Dipartimento di Matematica, Universita di Padova, I-35121 Padova, [email protected]

Abstract: We will describe some noteworthy ideals in categories of modules. Webegin with maximal ideals. Recall that a ring R is semilocal if R/J(R) is semisimpleartinian. A preadditive category is semilocal if it is non-null and the endomorphismring of every non-zero object is a semilocal ring. There are several natural examples ofsemilocal categories. Maximal ideals do not exist, in general, in arbitrary preadditivecategories, but do exist in semilocal categories. An additive functor F : A → B betweenpreadditive categories A and B is said to be a local functor if, for every morphismf : A → A′ in A, F (f) isomorphism in B implies f isomorphism in A. If C is asemilocal category, the canonical functor F : C → ⊕

M∈Max(C)C/M is a local functor.

Another ideal we will consider is the Jacobson radical of the category. The kernel ofany local functor F : A → B is an ideal of A contained in the Jacobson radical of A.If A is a preadditive category and I1, . . . , In are ideals of A, we will study when thecanonical functor A → A/I1 × · · · × A/In is a local functor. A weak form of theKrull-Schmidt theorem naturally appears in this setting.

We will conclude discussing Birkhoff’s Theorem (there exists a subdirect embedding ofA into a direct product of subdirectly irreducible preadditive categories) for skeletallysmall preadditive categories A and for the category A = Mod-R.

Keywords: Additive category, ideal (in a category)

2010 Mathematics Subject Classification: 16D90



1.7 Surender K. JAIN

Decomposition of Singular Matrices as Products ofIdempotents- A Survey

Surender K. JAIN

King Abdulaziz University Jeddah, SA and Ohio University, [email protected] , [email protected]

Abstract: J.M. Howie,in his paper, ”The subsemigroup generated by the idempotentsof a full transformation semigroup, J. London Math. Soc. 41 (1) (1966) 707-716”,made an important contribution on the subject dealing with semigroups generated byidempotents. Inspired by Howie, J. A. Erdos in his paper ”On products of idempotentmatrices, Glasgow Math Journal 8 (1967) 118-122” proved that singular matrices overa field can be presented as a product of idempotent matrices. This result of Erdoswas extended in different directions by several authors. For example, T. J. Laffey inhis paper ” Products of idempotent matrices, Linear and Multilinear Algebra 14 (4)1983 309-314” proved that every singular matrix over commutative Euclidean domainscan be decomposed as a product of idempotents matrices. K. P. S. Bhashara Rad,in his paper ”Products of idempotent matrices, Linear Algebra and its Applications430(2009) 2690-269” proved that for a commutative principal ideal domain the fol-lowing are equivalent: (1) Every square singular matrix is a product of idempotentmatrices, (2) every 2 × 2 matrix with second row zero is a product of idempotentmatrices, (3) every 2 × 2 matrix with second row zero and the g.c.d of the entries ofthe first row as 1, is a product of idempotent matrices J. Hannah and K.C. O’Meara,in their paper on ”Products of idempotents in regular rings, II, Journal of Algebra123, (1989), 223-239” showed that for a right self-injective regular ring R, an elementa is a product of idempotents if and only if R(r.ann(a)) = (l.ann(a))R= R(1- a)R.A. Alahmadi, S. K. Jain, and A. Leroy, in ”Decomposition of singular matrices intoidempotents, Linear and Multilinear Algebra (2013)” showed that if Ris a domain andit is one of the following types:(a) a commutative euclidean domain, (b) a local domainsuch that its radical J = Rg = gR with ∩Rgn = 0, (c) a commutative principal idealdomain with the property that every 2×2 singular matrix is a product of idempotentmatrices, or (d) a local right and left Bezout domain, then every singular matrix overRis a product of idempotent matrices

Alahmadi-Jain-Lam-Leroy have recently shown that if R is a right and left quasi-Euclidean domain and if A ∈ Mn(R) be such that l.ann A 6= 0,then A is a productof idempotent matrices whose entries can be described using continuant polynomials,generalizing Laffey’s result.



1.8 Sergio R. LOPEZ-PERMOUTH

Algebras Having Bases Consisting Entirely of Units

Sergio R. LOPEZ-PERMOUTH

Ohio University, Athens, [email protected]

Abstract: For an arbitrary ring R, we call an R-algebra A an invertible R- algebraif it has an R-basis consisting entirely of units in A. We will present various ideas onthe subject developed in an ongoing collaboration with Jeremy Moore, Nick Pilewskiand Steve Szabo. We will then conclude the talk by giving a progress report onthe problem of determining the directed graphs E for which the Leavitt path algebraLK(E) over a field K is an invertible K-algebra. This latter part is a joint project withNick Pilewski. We give examples of LPAs that are invertible and examples that arenot; we will also present some sufficient conditions for a Leavitt Path Algebra not tobe invertible. Locating the precise borderline between the two cases remains elusive.



1.9 W. Keith NICHOLSON

Unit Regular Modules

W. Keith NICHOLSON

University of Calgary, Calgary, [email protected]

Abstract: The unit regular elements in any module will be defined, and severalequivalent conditions will be discussed that generalize the properties of unit regularelements in a ring. A module is called unit regular if every element is unit regular, andtheorems of Camillo-Yu on exchange rings and Camillo-Khurana on clean rings willbe extended. The entire discussion is usefully phrased in terms of Morita contexts.



1.10 Barbara L. OSOFSKY

Compatible Ring Structures on Injective Hulls of F.E.Rings

Barbara L. OSOFSKY

Rutgers University, New Brunswick,[email protected]

Abstract: Let R be a ring with identity and injective hull ER = 〈E (RR) : 1 ∈ R 7→ 1R ∈ E〉

and set Λ = EndR (ER), Λ = Λ/J (Λ) . We study the question of when there is a ring

structure on E (R) which is compatible with right module multiplication by elementsof R if RR is finitely embedded (FE), i.e. RR is essential over a finitely generated socle.If ER is a rational extension of RR then such a ring structure is known to exist andbe unique. For nonrational extensions of FE rings R we use the map eval1 : Λ −→ E,eval1 (λ) = λ (1R) to define the ‘canonical’ injective hull ER = Λ /I and its restric-tion to a special submodule MR ⊆ E to characterize when E has a compatible ringstructure in the case that R is right perfect or commutative noetherian. The charac-terization depends only on the action of R on the left of the right socle of RR.



1.11 Edmund R.PUCZY LOWSKI

Applications of Modular Lattices in Some Studies ofModules and Rings

Edmund R.PUCZY LOWSKI

Institute of Mathematics, University of Warsaw, Warszawa, [email protected]

Abstract: Studying problems on modules or rings it is sometimes useful to put thestudies in a more general context of, say, specified categories or lattices. Constructionsavailable in that more general context can be used to establish some properties ofstudies objects and apply them to solve the original problems.

The aim of the talk is to present some old and new results on modules or rings obtainedby applying methods of modular lattices. They will mainly concern the Goldie, Krull-Rentschler and Gabriel dimensions.



1.12 Richard M. THOMAS

Representing Algebraic Structures Using Finite Automata

Richard M. THOMAS

Department of Computer Science, University of Leicester, Leicester LE1 7RH, [email protected]

Abstract: The notion of FA-presentability is motivated by an interest in possibleapproaches for understanding computability in algebraic structures (where a structureconsists of a set together with a finite collection of relations). A natural definitionwould be to take some general model of computation such as a Turing machine; astructure would then be said to be computable if its domain could represented by a setwhich can be recognized by a Turing machine and if there were decision-making Turingmachines for each of its relations. Notwithstanding this, there have been various ideasput forward to restrict the model of computation used; whilst the range of possiblestructures would decrease, the computation could become more efficient and certainproperties of the structures might become decidable.

One interesting approach was introduced by Khoussainov and Nerode [3] who con-sidered structures whose domain and relations can be checked by finite automata (asopposed to Turing machines); such a structure is said to be FA-presentable. This wasinspired, in part, by the theory of automatic groups introduced by Epstein et al [2] ;however, the definitions are somewhat different. One interesting point is that, whilstthe notion of automaticity in the sense of Epstein et al can be naturally generalizedto semigroups [1], it does not easily extend to arbitrary algebraic structures, whereasthe notion of FA-presentability does apply to all such structures.

We will survey some of what is known about FA-presentable structures, contrastingit with the theory of automatic groups and posing some open questions. The talkis intended to be self-contained, in that no prior knowledge of FA-presentability isassumed.

Keywords: Algebraic structures, FA-presentable, finite automaton, automatic groupsand semigroups

2010 Mathematics Subject Classification: 20F10; 68Q05; 68Q45



Reference:

[1] C. M. Campbell, E. F. Robertson, N. Ruskuc and R. M. Thomas, Automaticsemigroups, Theoretical Computer Science 365 (2001), 365–391.

[2] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. Levy, M. S. Paterson andW. Thurston, Word Processing in Groups (Jones and Barlett, 1992).

[3] B. Khoussainov and A. Nerode, Automatic presentations of structures, in D. Leivant(ed.), Logic and Computational Complexity (Lecture Notes in Computer Science960, Springer-Verlag, 1995), 367–392.



1.13 Robert WISBAUER

A Categorical View on Rings and Modules

Robert WISBAUER

Department of Mathematics, Heinrich Heine University, 40225 Dusseldorf, [email protected]

Abstract: To transfer techniques from module and ring theory to other areas ofmathematics it is of great advantage to understand the categorical content of a result,that is, to present a result by only using notions which are available in any category.Being aware of this one is also able to apply general category theory to modules andrings. Examples for this type of interchange are provided.


Contributed Speakers


2 Contributed Speakers

2.1 Jawad ABUHLAIL

Zariski Topologies for Coprime and Second Submodules

Jawad ABUHLAIL

Department of Mathematics and Statistics King Fahd University of PetroleumMinerals, Dhahran 31261, Saudi Arabia

[email protected]

Abstract: Let M be a non-zero module over an associative (not necessarily commuta-tive) ring. In this paper, we investigate the so-called second and coprime submodulesof M . Moreover, we topologize the spectrum Specc(M) of second submodules of Mand the spectrum Specc(M) of coprime submodules of M , study several properties ofthese spaces and investigate their interplay with the algebraic properties of M .

Keywords: dual Zariski topology, multiplication module, comultiplication module,co-prime submodule, second submodule

2010 Mathematics Subject Classification: 16D10, 13C05, 13C13, 54B99



2.2 Ahmad AL KHALAF

Relation between geometrically Lattice of subgroups andExchange property

Ahmad AL KHALAF, Abdullah ALJOUIEE

Department of Mathematics, Faculty of Science, Al-Imam Mohammad Ibn SuadIslamic University KSA

[email protected] and [email protected]

Abstract: A latticeL is called a semimodular if it satisfies the following condition: ifx and y both cover x ∧ y, then x ∨ y covers both x and y.

A latticeL is called an atomic if every x ∈ L is a join of atoms. A latticeL is called ageometrically lattice if it isatomic, semimodular, and does not contains infinite chain.

In this research we prove the following result:

Let G be a finite group and let sub G a lattice subgroups of a group G, then G is ageometrically lattice if and only if the generating operator of its subgroups satisfiesexchange property.

Keywords: geometrically lattice , exchange property.

Reference:

[1] Aigner M. Combinatorial theory, Springer-Verlag, Berlin- Heidelber New-York,(1989).

[2] Birkhoff G., Bartee T. Modern applied algebra, Mc Graw Hill box com. New York,London (1976).



2.3 Emine ALBAS

A Note on Generalized Derivations of Prime Rings

Emine ALBAS

Department of Mathematics, Ege University, 35100-Bornova, Izmir, [email protected]

Abstract: In this talk, R will represent an associative prime ring with center Z(R),Q its two sided Martindale quotient ring, U its Utumi quotient ring (sometimes Uis called the maximal ring of quotients), and C its extended centroid. An additivemapping d : R → R is called a derivation if

d(xy) = d(x)y + xd(y)

for all x, y ∈ R. There is often a tight connection between the structure of an associa-tive ring R and the behavior of some derivation defined on R.In [4], Wang and You proved that if R is a prime ring of characteristic 2 with a deriva-tion d and L a noncentral Lie ideal of R such that [d(u), u]n is central, for all u ∈ L,then R must satisfy S4 in 4 variables. The case where R is a semiprime ring wasalso examined by the authors. In [3], Lee showed that if R is a semiprime K-algebrawith unity where K is a commutative ring with 1, d is a non-zero derivation of R andf(x1, ..., xn) is a monic multilinear polynomial over K such that d(f(x1, ..., xn)) is zeroor invertible in R, then R is a division ring or the ring of all 2 × 2 matrices over adivision ring unless f(x1, ..., xn) is a central polynomial for R. Moreover in [2], it isproved that if R is a prime algebra over a commutative ring K with unity, f(x1, ..., xn)is a multilinear polynomial over K, d is a non-zero derivation on R such that for allx1, ..., xn in some non-zero ideal I of R, [d(f(x1, ..., xn)), f(x1, ..., xn)]k = 0, with k afixed positive integer, then either f(x1, ..., xn) is central-valued on R or charR = 2and R satisfies the standard identity S4 in 4 variables.

More recently in [1] V. De Filippis, O. M. Di Vincenzo have showed that if R is aprime ring of characteristic different from 2, f(x1, ..., xn) is a multilinear polynomialover a commutative ring K with unity and d is a non-zero derivation of R such that[d(f(x1, ..., xn)), f(x1, ..., xn)] is zero or invertible in R, for all x1, ..., xn ∈ R, theneither R is a division ring, or f(x1, ..., xn) is central-valued on R. An additive mapF : R → R is called a generalized derivation on R if there exists a derivation d on Rsuch that

F (xy) = F (x)y + xd(y)

for all x, y ∈ R. In this presentation our aim is to generalize the above results tothe case when the derivation d is replaced by a generalized derivation F of R and themultilinear polynomial f(x1, ..., xn) is replaced by the commutator [x, y] = xy − yx.We have the following:

Theorem. Let R be a prime ring of characteristic different from 2, with extendedcentroid C, F a non-zero generalized derivation of R, and I a non-zero two sided ideal



of R. Suppose that, for all x, y ∈ I, [F ([x, y]), [x, y]] is either zero or invertible in R.Then one of the following holds:

(i) F (x) = λx for all x ∈ R, where λ ∈ C;(ii) R = D, a division ring or R = M2(D) and for a suitable q ∈ U ,

F (x) = qx + xq for all x ∈ R;(iii) R satisfies the standard identity S4 and there exists q ∈ U such that

F (x) = qx + xq for all x ∈ R.

Keywords: prime rings, derivations, generalized derivations, Utumi quotient rings,two-sided Martindale quotient ring, differential identities

2010 Mathematics Subject Classification: 16N60, 16W25

Reference:

[1] De Filippis, V., Vincenzo, O. M.: Derivations on multilinear polynomials insemiprime rings, Comm. Algebra, 27(12), 5975-5983 (1999)

[2] Lee, P. H., Lee, T. K.: Derivations with Engel conditions on a multilinear poly-nomials Proc. Amer. Math. Soc., 124(9), 2625-2629 (1996)

[3] Lee, T. K.: Derivations with invertible values on a multilinear polynomialProc.Amer. Math. Soc., 119(4), 1077-1083 (1993)

[4] Wang, Y., You, H.: A note on commutators with power central values on Lieideals, Acta Math. Sinica, English Series, 22(6), 1715-1720 (2006)



2.4 Shakir ALI

On Commuting and Centralizing Derivations in Rings andAlgebras

Shakir ALI

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, [email protected]

Abstract: Let R be an associative ring(algebra) with center Z(R). For every asso-ciative ring R can be turned into a Lie ring(algebra) by introducing a new product[x, y] = xy − yx, known as Lie product. So we may regard R simultaneously as anassociative ring(algebra) and as a Lie ring(algebra). An additive mapping d : R → Ris said to be a derivation on R if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. Afunction f : R → R is called a centralizing on R if [f(x), x)] ∈ Z(R) holds for allx ∈ R. In the special case where [f(x), x)] = 0 for all x ∈ R, f is said to be commut-ing on R. The study of such mappings were initiated by E.C. Posner [ Proc. Amer.Math. Soc. 8(1957), 1093-1100]. In 1957, he proved that if a prime ring R has anonzero commuting derivation on R, then R is commutative. An analogous result forcentralizing automorphisms on prime rings was obtained by J.H. Mayne [Canad. J.Math. 19 (1976), 113-115].

In this talk, we will discuss the recent progress made on the topic and related areas.Moreover, some examples and counter examples will be discussed for questions raisednaturally.

Keywords: Prime ring, semi-prime ring, commuting mapping, centralizing mapping,derivation, generalized derivation.

2010 Mathematics Subject Classification: 16W25; 16R50; 16N60



2.5 Abdollah ALHEVAZ

On Graphical Representation of Skew Generalized PowerSeries Rings

Abdollah ALHEVAZ1 , Dariush KIANI1,2

1Department of Pure Mathematics, Faculty of Mathematics and Computer Sciences,Amirkabir University of Technology (Tehran Polytechnic), P.O. Box: 15875-4413,

Tehran, [email protected] and [email protected]

2School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box: 19395-5746, Tehran, Iran.

[email protected]

Abstract: Let R be any ring, (S,≤) a strictly (partially) ordered monoid andalso ω : S → End(R) a monoid homomorphism. A skew generalized power seriesring R[[S, ω,≤]] consists of all functions from a monoid S to a coefficient ring Rwhose support contains neither infinite descending chains nor infinite antichains, withpointwise addition, and with multiplication given by convolution twisted by an actionω of the monoid S on the ring R. There is considerable interest in studying if and howcertain graph-theoretic properties of rings are preserved under various ring-theoreticextensions. The zero-divisor graphs offer a graphical representation of rings so that wemay discover some new algebraic properties of rings that are hidden from the viewpointof classical ring theorists. In this paper, we examine the preservation of some graphinvariants of the zero-divisor graph of a non-commutative ring under extension to skewgeneralized power series ring.

Keywords: Skew generalized power series ring, graphical representation, graph in-variants.

2010 Mathematics Subject Classification: 05C25; 05C62; 06F05; 16S36.

Reference:

[1] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings,J. Algebra 296(2) (2006) 462-479.

[2] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring,J. Algebra 217(2) (1999) 434-447.

[3] M. Axtell, J. Coykendall and J. Stickles, Zero-divisor graphs of polynomials andpower-series over commutative rings, Comm. Algebra 33(6) (2005) 2043-2050.

[4] G. Marks, R. Mazurek and M. Ziembowski, A unified approach to various gener-alizations of Armendariz rings, Bull. Aust. Math. Soc. 81(3) (2010) 361-397.

[4] S.P. Redmond, The zero-divisor graph of a non-commutative ring, pp. 39-47 inCommutative Rings, edited by A. Badawi, Nova Sci. Publ., Hauppauge, NY,2002.



2.6 Mustafa ALKAN

On Graded Second and Coprimary Modules

Mustafa ALKAN , Secil CEKEN

Akdeniz University, Department of Mathematics, Antalya, [email protected]

Abstract: In this talk, we introduce and study the concepts of graded second (gr-second) and graded coprimary (gr-coprimary) modules which are different from secondand coprimary modules over arbitrary graded rings. We list some properties andcharacterizations of gr-second and gr-coprimary modules and also study graded primesubmodules of modules with gr-coprimary decompositions. We also deal with gradedsecondary representations for graded injective modules over commutative graded rings.By using the concept of σ-suspension (σ)M of a graded module M, we prove that agraded injective module over a commutative graded Noetherian ring has a gradedsecondary representation.



2.7 Abu Zaid ANSARI

Some Results on Additive Mappings With Involution inRings

Nadeem Ur REHMAN, Abu Zaid ANSARI

Department of Mathematics, Aligarh Muslim University, Aligarh-202002 [email protected] and [email protected]

Abstract: Let be semiprime ring. Let F : R → R is said to be a generalized Jordanderivation with associated Jordan derivation d : R → R such that F (x2) = F (x)x +xd(x) for all x ∈ R. A mapping d : R → Rd : R ! R is said to be (α, β)-derivation ifd(x2) = d(x)α(x) + β(x)d(x) for all x ∈ R,where α, β are the automorphisms of andF : R → R is said to be generalized Jordan (α, β)-derivation on R if there exists aJordan (α, β)-derivation d such that F (x2) = F (x)α(x) + β(x)d(x) for all x ∈ R. Inthis paper we extend some results in the setting of semiprime ∗-ring with some specialtype of additive mappings.



2.8 Orest D. ARTEMOVYCH

Associative Rings with Prime Lie Rings of Derivations

Orest D. ARTEMOVYCH

Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24,Cracow 31155 [email protected]

Abstract: Throughout, a ring R means an associative ring with 1. An additive mapd : R → R is called a derivation of R if d(ab) = d(a)b + ad(b) for all a, b ∈ R. The setDer(R) of all derivations of R is a Lie ring under operations of the pointwise addition“+” and the Lie multiplication “[−,−]”.

An ideal I of R is called δ-ideal if δ(I) ⊆ I, where δ ∈ Der(R). If R has no a δ-idealother than 0 and R, then it is said to be δ-simple. A ring is called prime (respectivelyδ-prime) if the product of any two nonzero ideals (respectively δ-ideals) is nonzero inR. Every δ-simple ring is δ-prime. Every prime ring is δ-prime for any δ ∈ Der(R).Recall also that a Lie ring D is called (Lie) prime if [A,B] 6= 0 for any two nonzeroideals A and B of D.

C. R. Jordan and D. A. Jordan (1978) have proved that if R is a prime 2-torsion freecommutative ring or a Noetherian δ-prime ring, then Rδ = rδ | r ∈ R and [Rδ,Rδ]are prime Lie rings.

A. Nowicki (1985) have extended these results.

M. A. Chebotar and P.-H. Lee (2006) have shown that if R is a reduced (i.e., withoutnonzero nilpotent elements) 2-torsion free commutative D-prime ring, D is a nonzeroLie subring and an R-submodule of Der(R), then D is a prime Lie ring.

P.-H. Lee and C.-K. Liu (2007) have establish that if R is a 2-torsion free commutativeD-prime ring, D is a Lie subring and an R-submodule of Der(R), then any nonzeroLie ideal A of D is a prime Lie ring.

Let C = RD be the ring of D-constants of R, C∗ the set of nonzero elements of C, Kthe field of fractions of C and S = C−1R the localization of R at C∗. Each derivationdelta of R in D extends uniquely to a derivation δ of S. Let D = δ | δ ∈ D andD = KD. C.-H. Liu and D. S. Passman (2007) have proved that if R is a commutativeD-prime ring, D is a nonzero Lie subring and an R-submodule of Der(R), then

(a) if charR = 2 and D = Rδ 6= 0, then D is Lie prime if and only if δ(δ(R)δ(R)) 6=0,

(b) D is Lie prime except possibly when char(R) = 2 and D is cyclic as S-module,

(c) if charR = 2 and D = Sθ is cyclic as an S-module, then D is Lie prime if andonly if θ(θ(R)θ(R)) 6= 0,

(d) D is Lie prime if and only if char(R) 6= 2 or dimKS 6= 2,

(e) if D is not Lie prime, then D is the unique nonabelian Lie algebra of K-dimension 2.



A Lie ring D is called (Lie) semiprime if, for any nonzero Lie ideal J of D, we have[J, J ] 6= 0. C.-H. Liu (2006) has proved that if R is a 2-torsion free commutativeD-semiprime ring, D is a Lie subring and an R-submodule of Der(R), then D is Liesemiprime.

We plane to present new results on (not necessary commutative) rings R with theprime Lie ring Der(R) of derivations.



2.9 Umamaheswaran ARUNACHALAM

Strongly FFI-Injective Module and Its Dimension

Umamaheswaran ARUNACHALAM

Department of Mathematics, Periyar University Salem, Tamil Nadu [email protected]

Abstract:



2.10 Mohammad ASHRAF

Permuting n-derivations in Prime Near-Rings

Mohammad ASHRAF

Department of Mathematics, Aligarh Muslim University, Aligarh, [email protected]

Abstract: N be a zero-symmetric near-ring. A map D : N ×N × · · · ×N︸︷︷︸

n−times

−→ N is

said to be permuting if the equation D(x1, x2, · · · , xn) = D(xπ(1), xπ(2), · · · , xπ(n)) holdsfor all x1, x2, · · · , xn ∈ N and for every permutation π ∈ Sn, where Sn is the permu-tation group on 1, 2, · · · , n. A map d : N → N defined by d(x) = D(x, x, · · · , x)for all x ∈ N where D : N ×N × · · · ×N︸︷︷︸

n−times

→ N is a permuting map, is called the

trace of D. A permuting n-additive(i.e., additive in each argument) mapping D :

N ×N × · · · ×N︸︷︷︸n−times

−→ N is called a permuting n-derivation if D(x1x′

1, x2, · · · , xn) =

D(x1, x2, · · · , xn)x′

1 +x1D(x′

1, x2, · · · , xn) holds for all x1, x′

1, · · · , xn ∈ N . Of course,a permuting 1- derivation is a derivation and permuting 2-derivation is a symmet-ric bi-derivation.The concepts of symmetric bi-derivation and permuting n-derivationhave already been introduced in rings by G. Maksa, [ On the trace of symmetricbi-derivations, C. R. Math. Rep. Sci. Canada,9(1987), 303-307] and Park, K.H.and Jung, Y.S., [ On permuting 3-derivations and commutativity in prime near-rings,Commun. Korean Math. Soc. 25 , (2010), 1-9] respectively. In the present talk, moti-vated by these concepts, we define permuting n-derivations and generalized permutingn-derivations in near-rings, and investigate commutativity of addition and multiplica-tion of N . Further, under certain constraints on a n!-torsion free prime near-ring N ,it is shown that a permuting n-additive mapping D on N is zero if the trace d of D iszero. Finally, some more related results are also obtained.

Keywords: Left near-rings , zero symmetric , derivations , permuting n-derivations

2010 Mathematics Subject Classification: 16W25 , 16Y30



2.11 Yıldız AYDIN

Frattini-Supplemented Groups

Yıldız AYDIN , Ali PANCAR

Department of Mathematics, Ondokuz Mayıs University, Atakum, Samsun [email protected] and [email protected]

Abstract: In this study Frattini-supplement subgroup and Frattini-supplementedgroup are defined by Frattini subgroup in group theory as a correspondence to Rad-supplemented module in module theory. By these definations its shown that abeliangroups are satisfiying this property and every conjugate of a Frattini-supplement sub-group is also a Frattini-supplement. A group action of a group is defined over the setof Frattini-supplements of a normal subgroup of the group by conjugation and withthis action some useful consequences are obtained of Frattini-supplemented groups.

2010 Mathematics Subject Classification: 58E40,58D19,20D25

Reference:

[1] Robinson,D.J.S. A course in the theory of groups. Springer-Verlag New York(1982).

[2] Hausen, J. Supplemented nilpotent groups Rendiconti del Seminario Matematicodella Universita di Padova 65 (1981), 35-46.

[3] Dixon, J.D., Mortimer, B. Permutation groups. Springer-Verlag New York (1996).

[4] Otal,J., The Frattini Subgroup of A Group. Margarita Mathematica en Memoriade Jose Javier (Chicho) Guadalupe Hernandez, Spain (2001)

[5] Beidleman, J.C., Seo, T.K., Generalized Frattini Subgroup of Finite Groups,Pasific Journal of Mathematics, 23 (1967),no.3,441-450.



2.12 Ayman BADAWI

2-Absorbing Primary Ideals in Commutative Rings

Ayman BADAWI1 , Unsal TEKIR 2 and Ece YETKIN 2

1 Department of Mathematics, American University of Sharjah, Sharjah, UnitedArab Emirates

[email protected]

2 Marmara University,Department of Mathematics, Istanbul, [email protected] and [email protected]

Abstract: In this paper, we study 2-absorbing primary ideals, which are a gen-eralization of primary ideals. A proper ideal Q of a commutative ring R is calleda 2-absorbing primary ideal of R if whenever a, b, c ∈ R with abc ∈ Q, then eitherab ∈ Q or ac ∈ √

Q or bc ∈ √Q. It is shown that if I is an ideal of R such that

√I

is a prime ideal of R, then I is a 2-absorbing primary ideal of R. It is shown that ifQ is a 2-absorbing primary ideal of R, then

√Q is a 2-absorbing ideal of R in the

sense of Badawi [1] and [2]. If R is a principal ideal domain, then we characterize all2-absorbing primary ideals of R; also a characterization of Dedekind domains in termsof 2-absorbing primary ideals is established.

Keywords: 2-absorbing ideals, n-absorbing ideals, primary and prime ideals, 2-absorbing primary ideals.

2010 Mathematics Subject Classification: 13A15; Secondary 13F05, 13G05.

Reference:

[1] Badawi, A. On 2-absorbing ideals of commutative rings, Bull. Austral. Math.Soc., 75(2007) 417-429.

[2] Anderson, D.F., Badawi A. On n-absorbing ideals of commutative rings, Commu-nications in Algebra, 39 (2011), 1646-1672.



2.13 Evgenii BASHKIROV

On Linear Groups over a Finitely GeneratedCommutative Algebra

Evgenii BASHKIROV

Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul,Turkey

[email protected]

Abstract: The following result is proved.

Theorem. Let k be a field of characteristic 6= 2, K a commutative finite-dimensionalk-algebra generated by the elements α1, α2, . . . , αp(p ≥ 1). Let di be the degree of apolynomial over k whose root is αi and d = d1d2 . . . dp. There exists a positive integerC depending on d only (that, is C does on depend on α) with the following property: ifk > 2C + 1, n is integer, n ≥ 3, X is a subgroup of GLn(K) containing SLn(k) and

X is generated by transvections, then X is isomorphic to the extension of a suitablelocally nilpotent group by the group SLn(L), where L is a semi-simple homomorphicimage of a certain subalgebra of K. If, moreover, the quotient-algebra of K by itsJacobson radical is separable, then L is isomorphic to a semi-simple subalgebra of K.

Reference:

[1] Bashkirov, E. L., Gupta C. K. Linear groups over integral extensions of semilocalcommutative rings. Appears in Communications in Algebra.



2.14 Samruam BAUPRADIST

General form of Slightly Compressible Modules

Phatsarapa Janmuang1, Samruam BAUPRADIST1 and Ronnason Chinram2

1 Department of Mathematics and Computer Science, Faculty of Science,Chulalongkorn University, Bangkok 10330, [email protected] and [email protected]

2 Department of Mathematics and Statistics, Faculty of Science, Prince of SongklaUniversity, Hat Yai, Songkla 90110, Thailand

[email protected]

Abstract: This paper is concerned with a general form of slightly compressible mod-ules. The concept of a general form of slightly compressible modules will be calledM -slightly compressible modules where M is a right R-module, following this, a rightR-module N is called an M -slightly compressible module if for every non-zero sub-module A of N there exists a non-zero homomorphism from M to A. We give somecharacterizations and properties of M -slightly compressible module.

Keywords: slightly compressible modules, M -slightly compressible modules, M -cyclic compressibly injective modules.

2010 Mathematics Subject Classification: 16D10;16D50;16D70

Reference:

[1] Anderson, F. W., Fuller, K. R. Rings and Categories of Modules. 2nd ed. GraduateTexts in Mathematics New York: Springer-Verlag. (1974) No.13.

[2] Baupradist, S., Asawasamrit, S. On Fully-M-Cyclic Modules. J. Math. Res.(2011) 3(2) : 23-26.

[3] Kasch, F. Modules and Rings. London Math. Soc. Monographs (1982) 17(C.U.P.).

[4] Pandeya, B. M., Chaturvedi, A. K., Gupta, A.J. Applications of Epi-retractableModules. Bull. IranianMath. Soc. 1 (2012) : article in

[5] Sanh, N. V., Shum, K. P., Dhompongsa, S., Wongwai, S. On Quasi-principallyInjective Modules. Algebra Colloq. (1999) 6 (3) : 269-276.

[6] Smith, P. F. Modules with many homomorphisms. J. Pure Appl. Algebra (2005)197 : 305-321.



2.15 Firdhousi BEGUM

Ulm Supports on QTAG-Modules

Firdhousi BEGUM 1, Ayazul HASAN2

1 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, Indiafirdousi [email protected]

2 Department of Mathematics,Integral University, Lucknow, [email protected]

Abstract: A right module M over an associative ring with unity is a QTAG-moduleif every finitely generated submodule of any homomorphic image of M is a direct sumof uniserial modules. Mehdi and Naji introduced the notion of transitivity for QTAG-modules. Motivated by the transitivity and full transitivity we study full transitivepairs of QTAG-modules and obtain several characterizations. Here we examine howthe formation of direct sums of QTAG-modules affects transitivity and full transitivity.We extend this concept by defining Ulm supports of QTAG-modules and consequentlyderive more results about the interrelationships of the various transitivities.

Keywords: QTAG-modules, transitive modules, Ulm invariants.

2010 Mathematics Subject Classification: 16K20



2.16 Vijay Kumar BHAT

Completely generalized right primary rings and theirextensions

Vijay Kumar BHAT

School of Mathematics, SMVD University, Katra, [email protected]

Abstract: A ring R is said to be a completely generalized right primary ring (c.g.r.pring) if a, b ∈ R with ab = 0 implies that a = 0 or b is nilpotent.

Let now R be a ring and σ an automorphism of R. In this paper we extend theproperty of a completely generalized right primary ring (c.g.r.p ring) to the skewpolynomial ring R[x;σ]. We say that R[x;σ] is an extended completely generalizedright primary ring (e.c.g.r.p ring) if for f(x), g(x) ∈ R[x;σ] (say f(x) =

∑n

i=0 xiai and

g(x) =∑m

j=0 xjbj), f(x)g(x) = 0 implies that f(x) = 0 or bj is nilpotent for all j,

0 ≤ j ≤ m.

With this we prove that if R is a c.g.r.p ring and σ an automorphism of R, then R[x;σ]is an extended c.g.r.p ring.

Keywords: Ore extension, automorphism, derivation, completely prime ideal

2010 Mathematics Subject Classification: 16N40, 16P40, 16S36

Reference:

[1] V. K. Bhat, A note on completely prime ideals of Ore extensions, Internat. J.Algebra Comput., Vol. 20(3) (2010), 457-463.

[2] C. Gorton and H. Heatherly, Generalized primary rings, Mathematica Pannonica,17(1) (2006), 17-28.

[3] C. Gorton, H. E. Heatherly and R. P. Tucci, Generalized primary rings, Interna-tional Electronic Journal of Algebra, 12(2012) 116-132.



2.17 Ismail Naci CANGUL

On Some Trigonometric Minimal Polynomials1

Ismail Naci CANGUL

Uludag University, Faculty of Arts and Science, Department of Mathematics, 16059Bursa, TURKEY

[email protected]

Abstract: In this talk, the author will give a survey of several trigonometric minimalpolynomials of 2 cos (π/n) , cos (π/n) and cos (2π/n) over rationals using Dickson andChebycheff polynomials.

2010 Mathematics Subject Classification: 12E05, 33B10

Reference:

[1] Cangul, I. N., Normal Subgroups of Hecke Groups, PhD Thesis, SouthamptonUniversity, 1994

[2] Cangul, I. N., The Minimal Polynomial of cos(2π/n) over Q, Probl. Mat. Bydg.15 (1997), 57-62, Probl. Mat. Bydg. 15 (1997), 57-62Probl. Mat. Bydg. 15(1997), 57-62

[3] Demirci, M. & Cangul, I. N., The Constant Term of The Minimal Polynomial ofcos(2π/n) over Q, Fixed Point Theory and Applications, 77 (2013), 1-8

[4] Lehmer, D. H., A Note on Trigonometric Algebraic Numbers, Amer. Math.Monthly, 40 (1993), 165-166

[5] Watkins, W. & Zeitlin, J., The Minimal Polynomial of cos(2π/n), Amer. Math.Monthly, 100(5) (1993), 471-474

[6] Bayad, A., Cangul, I. N., The Minimal Polynomial of 2 cos(π/q) and DicksonPolynomials, Applied Mathematics and Computation, 218 (2012), 7014–7022 1

1Acknowledgement: The author is supported by the Commission of Scientific ResearchProjects of Uludag University, project numbers 2012/15, 2012/19 and 2012/20.



2.18 Paula A.A.B. CARVALHO

Injective Hulls of Simple Modules over DifferentialOperator Rings

Paula A.A.B. CARVALHO, Can HATIPOGLU and Christian LOMP

Centro de Matematica, Universidade do Porto, Rua Campo Alegre 687, 4169-007Porto, [email protected]

Abstract: The Jacobson’s conjecture is an open problem in ring theory and askswhether the intersection of the powers of the Jacobson radical of a two-sided Noethe-rian ring is zero. Jategaonkar answered the conjecture in the affirmative for a Noethe-rian ring R under an additional assumption (called FBN) which in particular impliesthat any finitely generated essential extension of a simple left R-module is Artinian.The latter condition, denoted by (⋄), is a sufficient condition for a positive answer tothe Jacobson’s conjecture.

We will consider property (⋄) for differential operator rings R[x; d] with R a com-mutative Noetherian ring and d a derivation. As a consequence we characterize Oreextensions S = K[x][y;σ, d] satisfying the above property.

Keywords: Noetherian rings, injective modules, essential extensions of simple mod-ules, Artinian modules, differential operator rings

2010 Mathematics Subject Classification: 16E70; 16P40



2.19 Canan Celep YUCEL

A Note on ECS-Modules

Canan Celep YUCEL

Pamukkale University, Faculty of Arts and Science, Department of Mathematics,Denizli, [email protected]

Abstract: A module M is said to satisfy the ECS condition if every ec-closed submod-uleof M is a direct summand. It is known that the class of ECS-modules is not closedunder direct sums. In this paper, we studied when a direct sum of two modules is anECS-module and when an ECS-module has a decomposition into uniform submodules.



2.20 Jianlong CHEN

Coherence, (m,n)-injectivity and Generalized MorphicProperty of Rings

Jianlong CHEN

Department of Mathematics, Southeast University, Nanjing, [email protected]

Abstract: A ring R is left coherent if each finitely generated left ideal of R is finitelypresented. For two positive integers m and n, R is said to be left (m,n)-injective ifevery left R-homomorphism from an n-generated submodule of RR

m to RR extendsto one from RR

m to RR. R is called left generalized morphic if for every element ain R, l(a) = Rb for some b ∈ R, where l(a) denotes the left annihilator of a in R. Inthis talk, we discuss the coherence, the (m,n)-injectivity and the generalized morphicproperty of the upper triangular matrix ring Tk(R) and its subring (up to isomorphic)R[x]/(xk) (k ≥ 1). It is proved that R is left coherent if and only if Tk(R) is leftcoherent if and only if R[x]/(xk) is left coherent (k ≥ 1). Moreover, it is shown thatR is left coherent and left Bezout if and only if Tk(R) is left generalized morphic foreach k ≥ 1. Furthermore, various sufficient and necessary conditions are obtained forR[x]/(x2) to be left (m,n)-injective, for R[x]/(xk) (k > 2) to be left P -injective andfor Tk(R) to be left generalized morphic. ( This is a joint work with Qiongling Liu.)

Keywords: Coherent ring, (m,n)-injectivity, Generalized morphic ring, Bezout ring,Upper triangular matrix ring.

2010 Mathematics Subject Classification: 16D50; 16P70; 16S50

Reference:

[1] Chen, J. L., Ding, N. Q., Li, Y. L., Zhou, Y. Q. On (m,n)-injectivity of modules.Comm. Algebra 29 (2001), 5589–5603.

[2] Chen, J. L., Zhou, Y. Q. Extensions of injectivity and coherent rings. Comm.Algebra 34 (2006), 275–288.

[3] Lee, T.-K., Zhou, Y. Q. Regularity and morphic property of rings. J. Algebra 322(2009), 1072–1085.

[4] Nicholson, W. K., Sanchez Campos, E. Rings with the dual of the isomorphismtheorem. J. Algebra 271 (2004), 391–406.

[5] Zhu, H. Y., Ding, N. Q. Generalized morphic rings and their applications. Comm.Algebra 35 (2007), 2820–2837.



2.21 Jules Clement MBA

Groups from module actions

Jules Clement MBA

Department of Mathematics, University of Johannesburg, Kingsway Campus, P.O.Box 524, Auckland Park 2006, South Africa

[email protected]

Abstract: In [Hilton, P. Non-cancellation properties for certain finitely presentedgroups. Classical and categorical algebra (Durban, 1985). Quaestiones Math. 9 (1986),no. 1-4, 281-292], a description of the non-cancellation set of a finitely presented groupis given. We generalize this to the case of R-modules, where R is some commutativering. To achieve this, we form a category of metabelian groups CR whose objects G aresemidirect products obtained from R-modules. For an object G, a group structure isinduced on the non-cancellation set of the localization of G at the set of primes 3, 7. An example of trivial non-cancellation set in CR is given.

Keywords: Localization, Non-cancellation, Semi-direct product.

2010 Mathematics Subject Classification: 20F06;20E99;13B30

Reference:

[1] Hilton, P. Non-cancellation properties for certain finitely presented groups. Clas-sical and categorical algebra (Durban, 1985). Quaestiones Math. 9 (1986), no.1-4, 281-292.

[2] Scevenels, D. On the Mislin genus of a certain class of nilpotent groups. Comm.Algebra. 26 (1998), no. 5, 1367-1376.

[3] Scevenels, D., Witbooi, P.J. Non-cancellation and Mislin genus of certain groupsand H0-spaces. Journal of Pure and Applied Algebra. 170 (2002), 309-320.

[4] Warfield, R. Genus and cancellation for groups with finite commutator subgroup.J. Pure Appl. Algebra. 6 (1975) 125-132.

[5] Webb, P. J. The minimal relation modules of a finite abelian group. J. Pure Appl.Algebra. 21 (1981), no. 2, 205-232.

[6] Witbooi, P. J. Non-cancellation for certain classes of groups. Comm. Algebra 27(1999), no. 8, 3639-3646.

[7] Witbooi, P. J. Generalizing the Hilton-Mislin genus group. J. Algebra 239 (2001),no. 1, 327-339.



2.22 Secil CEKEN

On Second Submodules and the Second Radical ofModules

Secil CEKEN1, Mustafa ALKAN1 and Patrick F. SMITH2

1 Department of Mathematics, Akdeniz University, Antalya, [email protected] and [email protected]

2 Department of Mathematics, University of Glasgow, Glasgow, [email protected]

Abstract: Let R be a ring with identity and M be a unital right R-module. Anonzero submodule N of M is called a second submodule if N and all its nonzerohomomorphic images have the same annihilator in R. The second radical of a moduleM is defined to be the sum of all second submodules of M . In this talk we give somecharacterizations of second modules over noncommutative rings. We give some generalproperties of the second radical and determine the second radical of some modules.The second radical of a module M is closely related to the socle of M . We investigatewhen the second radical of a module M is equal to the socle of M . We also give somecharacterizations of rings by using the concept of the second radical of a module.

Keywords: Second submodule, second radical, socle of a module.

2010 Mathematics Subject Classification: 16D50,16D60, 16D80, 16N20, 16N60.

Reference:

[1] Abuhlail, J. Zariski topologies for coprime and second submodules, Algebra Col-loquium, to appear.

[2] Annin, S. Attached primes over noncommutative rings, J. Pure Appl. Algebra,212 (3), (2008), 510-521.

[3] Ansari-Toroghy, H., Farshadifar, F. On the dual notion of prime submodules,Algebra Colloq., 19 (1), (2012), 1109-1116.

[4] Ansari-Toroghy, H., Farshadifar, F. On the dual notion of prime submodules II,Mediterr. J. Math., 9 (2), (2012) 327-336.

[5] Ceken, S., Alkan, M. Dual of Zariski topology for modules, Book Series: AIPConference Proceedings, 1389 (1), (2011), 357-360.

[6] Ceken, S., Alkan, M., Smith, P. F. Second modules over noncommutative rings,Communications in Algebra, 41 (1), (2013), 83-98.

[7] Ceken, S., Alkan, M. On graded second and coprimary modules and graded sec-ondary representations, Bulletin of the Malaysian Mathematical Sciences Soci-ety, to appear.

[8] Ceken, S., Alkan, M., Smith, P. F. The dual notion of the prime radical of amodule, submitted.



2.23 Nadeem Ahmad DAR

On ∗-commuting Mappings and Derivations in Rings withInvolution

Nadeem Ahmad DAR, Shakir ALI

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, [email protected] and [email protected]

Abstract: Let R be a prime ring with involution ∗. An additive mapping f : R → Ris said to be ∗-centralizing on R if [f(x), x∗] ∈ Z(R) holds for x ∈ R; in the specialcase when [f(x), x∗] = 0 holds for all x ∈ R, the mapping f is said to be ∗-commutingon R. The main purpose of this paper is describe the structure of a pair of additivemappings which is ∗-commuting on a semi-prime ring with involution. Moreover wealso prove the following result: Let R be a prime ring with involution ∗ such thatchar(R) 6= 2. Let d be a nonzero derivation of R such that d([x, x∗]) ± [x, x∗] = 0 forall x ∈ R, then R satisfies S4, the standard polynomial identity of degree 4. Moreoverif we assume (S(R) ∩ Z(R)) 6= 0. Then R is commutative.

Keywords: Prime ring, semiprime ring, normal ring, involution, derivation.



2.24 Kinkar Ch. DAS

Sharp Bounds for Energy of Graphs

Kinkar Ch. DAS

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic ofKorea

[email protected]

Abstract: The energy of a graph G, denoted by E(G), is defined as the sum of theabsolute values of all eigenvalues of G . In this paper we present some lower and upperbounds for E(G) in terms of number of vertices, number of edges and determinantof the adjacency matrix of graph G. Moreover, we compare our lower bound withpreviously known lower bounds.

Keywords: Graph, Spectral radius, Energy, Determinant

2010 Mathematics Subject Classification: 05C50

Reference:

[1] D. M. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs–Theory and Application,Academic Press, New York, 1980.

[2] K. C. Das, S. A. Mojallal, Upper bounds for the energy of graphs, MATCH Com-mun. Math. Comput. Chem. 70 (2) (2013) 657–662.

[3] K. C. Das, S. A. Mojallal, I. Gutman, Improving McClelland? lower bound forenergy, MATCH Commun. Math. Comput. Chem. 70 (2) (2013) 663–668.

[4] I. Gutman, Bounds for total π-electron energy of polymethines, Chem. Phys.Lett. 50 (1977) 488-?90.

[5] J. H. Koolen, V. Moulton, Maximal energy graphs, Adv. Appl. Math. 26 (2001)47–52.

[6] J. Li, X. Li, Y. Shi, On the maximal energy tree with two maximum degree vertices,Linear Algebra Appl. 435 (2011) 2272?2284.

[7] X. Li, Y. Shi, I. Gutman, Graph energy, Springer, New York, 2012.

[8] B. J. McClelland, Properties of the latent roots of a matrix: The estimation ofπ-electron energies, J. Chem. Phys. 54 (1971) 640–643.

[9] B. Zhou, Energy of graphs, MATCH Commun. Math. Comput. Chem. 51 (2004)111–118.



2.25 Ali Ahmed DAW

On Fuzzy Sets Forming a Boolean Algebra

Ali Ahmed DAW

Tripoli University, Department of Mathematics Faculty of Science, P.O.Box 13211,Tripoli, Libya

A A [email protected]

Abstract: In the family IX of fuzzy subsets of a set X we introduce an order relationand the operations: probabilistic sum,product and complementation. Then we givenecessary and sufficient conditions for some A⊆IX to be a Boolean algebra. By thiswe establish connection between fuzz sets and Boolean algebras, which can be usefulto study both of them.

Reference:

[1] Dubois, D Prade H, Fuzzy sets and systems. New York, 1980.

[2] Filep, L; Studies in fuzzy relations using triangular norms. Information Sciences(to appear).

[3] Maeda, F Maeda, S; Theory of symmetric lattices. Berlin, 1971.

[4] Zadeh, L.A. : Fuzzy sets. Information and Control. 8(1965), 338-353.

[5] Daw, A.A. A formula for the number of retracts of finite Boolean Algebra. Demon-stration Math. Vol. XXII, No. 3, 1989.



2.26 Fatemeh DEHGHANI ZADEH

Graded Local Cohomology Modules with Respect to aPair of Ideals

Fatemeh DEHGHANI ZADEH

Department of Mathematics, Islamic Azad University, Yazd Branch, Yazd, [email protected]

Abstract: LetR = ⊕n≥0Rn be a graded commutative Noetherian ring,where the basering R0 is a commutative Noetherian local ring with maximal ideal m0. Moreover, weuse I0, J0 that is denoted proper ideals of R0 and we set R+ = ⊕n>0Rn for theirrelevant ideal of R, I = I0 + R+, J = J0 + R+, and m = m0 + R+. Also, we useM = ⊕n∈ZMn to denote a non-zero, finitely generated graded R-modules. We showthat, the i-th generalized local cohomology module Hi

I,J(M) inherits natural gradingfor each i ∈ N0 (where N0 denotes the set of all non-negative integers). We use thenotation Hi

I,J(M)n to denote the n-th graded component of HiI,J (M) for each n ∈ Z

(where Z denotes the set of integers).

In this paper, the behaviour of the n-th graded component HiI,J(M)n of the local

cohomology module with respect to a pair of ideals (I, J) as n −→ −∞ is investigated.

Keywords: Local cohomology, Artinian module, Cofinite module.

2010 Mathematics Subject Classification: 13D45, 13E05, 14B15

Reference:

[1] M. P. Brodmann, R. Y. Sharp, Local cohomology: An Algebraic introduction withgeometric applications, Cambridge Univ. Press 1998.

[2] F. Dehghani-Zadeh, Finiteness properties generalized local cohomology with re-spect to an ideal containing the irrlevant ideal, J, Korean. Soc. 49 (2012), No.6,1215-1227.

[3] F. Dehghani-Zadeh, On the finiteness properties of generalized local cohomologymodules, International Electronic Journal of Algebra, 10 (2011), 113-122.

[4] F. Dehghani-Zadeh, Fiter regular sequence and generalized local cohomology withrespect to a pair of ideals, To appear in Journal of Mathematical Extension.

[5] A. Grothendieck, Cohomological local des faisceaur coherents et theoremes deLefschetz locaux et globaus (SGA2), North-Holland, Amsterdam, 1968.

[6] R. Hartshorne, , Affine duality and cofiniteness ,, Invent. Math, (1970), 145-164.

[7] C. Hunkek, Problems on local cohomology, in: Free resolutions in commutativealgebra and algebraic geometry, Research Notes In Math., ed, by Eisenbud. andHuneke C., Jones and Bartlett, Boston, 1992, 93-108.

[8] L. Melkersson,, On asymptotic stability for sets of prime ideals connected with thepowers of an ideal,, Math. Proc. Cambridge philos. Soc. 107 (1990) 267-271.



[9] L. Melkersson, Properties of cofinite modules and applications to local cohomology,Math. Proc. Cambridge Phil. Soc. 125(1999), 417-423.

[10] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005),649-668.

[11] R. Takahashi, Y. Yoshino, T. Yoshinawa, Local cohomology based on a nonclosedsupport defined by a pair of ideals, J. Pure Apple. Algebra 213 (2009) 582-600.

[12] W. V. Vasconcelos, Divisor Theory in Module Categories, North-Holland Math-ematics Studies, 14, Elsevier, 1974.



2.27 Cagri DEMIR

On a Triplet of Generalized Derivations of Prime Rings onMultilinear Polynomials

Cagri DEMIR1, Giovanni SCUDO2

1Department of Mathematics, Ege University, Bornova, Izmir, [email protected]

2Department of Mathematics, University of Messina, 98166 Messina, [email protected]

Abstract: Let R be a ring. An additive map d : R → R satisfying d(xy) = d(x)y +xd(y) for all x, y ∈ R is called a derivation of R. In [5], Bell and Kappe showed thata prime ring does not possess any nonzero derivation which acts as a homomorphismor as an anti-homomorphism on a nonzero left ideal of the ring. Since then, by themotivation of this result, researchers extensively studied similar kinds of problems forseveral types of mappings of (semi)prime rings and, extended the result of Bell andKappe in various directions (see, e.g., [1-3], [5], [9-12], [14], [15] and references therein).

In my talk, I will announce a more involved progress in this vein, and if time permits,I will mention some related open problems. More precisely, if R is a prime ringof characteristic different from 2, C is its extended centroid and f(X1, . . . , Xn) is anoncentral multilinear polynomial over C, then we characterize generalized derivationsF , G and H of R for which F (x)G(y) = H(xy) for all x, y ∈ f(R), where f(R) is theset of all evaluations of f on R. As an application, we show that the only nonzerogeneralized derivation of a prime ring R of characteristic different from 2 which actsas an homomorphism on a noncentral Lie ideal of R is the identity map.

Keywords: Prime ring, generalized derivation, homomorphism, generalized polyno-mial identity.

2010 Mathematics Subject Classification: 16N60;16W20;16W25

Reference:

[1] Ali, S.; Huang, S., On generalized Jordan (α,β)-derivations that act as homomor-phisms or anti-homomorphisms. J. Algebra Comput. Appl. 1 (2011), no. 1,13?19.

[2] Ali, A., Kumar, D., Generalized derivations as homomorphisms or as anti-homomorphisms in a prime ring. Hacet. J. Math. Stat. 38 (2009), no. 1,17?20.

[3] Ali, S., On generalized left derivations in rings and Banach algebras. AequationesMath. 81 (2011), no. 3, 209–226.

[4] K.I. Beidar, W.S. Martindale III, V. Mikhalev, Rings with generalized identities.Pure and Applied Math., Dekker, New York, (1996).



[5] Bell, H. E., Kappe, L. C. Rings in which derivations satisfy certain algebraicconditions. Acta Math. Hungar. 53 (1989), no. 3-4, 339?346.

[6] C.M. Chang and T.K. Lee, Additive subgroups generated by polynomial valueson right ideals. Comm. Algebra 29 (2001), no.7, 2977–2984.

[7] C.L. Chuang, GPIs having coefficients in Utumi quotient rings. Proc. Amer.Math. Soc. 103 (1988), no. 3, 723–728.

[8] C.L. Chuang and T.K. Lee, Finite products of derivations in prime rings. Comm.Algebra 30 (2002), no. 5, 2183–2190.

[9] De Filippis, V., Generalized derivations as Jordan homomorphisms on Lie idealsand right ideals. Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 12, 1965?1974.

[10] Dhara, B., Generalized derivations acting as a homomorphism or anti-homomorphismin semiprime rings. Beitr. Algebra Geom. 53 (2012), no. 1, 203?209.

[11] Eremita, D., Ilisevic, D., On (anti)-multiplicative generalized derivations. Glas.Mat. Ser. III 47(67) (2012), no. 1, 105–118.

[12] Golbasi, O., Koc, E., Notes on generalized (σ,τ)-derivations. Rend. Semin. Mat.Univ. Padova 123 (2010), 131–139.

[13] N. Jacobson, Structure of rings. American Mathematical Society, ColloquiumPublications, vol. 37. American Mathematical Society, 190 Hope Street, Prov.,R. I., (1956).

[14] Wang, Y., You, H. Derivations as homomorphisms or anti-homomorphisms onLie ideals. Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 6, 1149?1152.

[15] Xu, X.-W., Zhang, H.-Y., A note on generalized left (θ,φ)-derivations in primerings. Hacet. J. Math. Stat. 40 (2011), no. 4, 523?529.



2.28 Nanqing DING

Duality Pairs Induced by Gorenstein Projective Moduleswith Respect to Semidualizing Modules

Nanqing DING

Department of Mathematics, Nanjing University, Nanjing 210093, [email protected]

Abstract: Let C be a semidualizing module over a commutative Noetherian ring R.We investigate duality pairs induced by C-Gorenstein projective modules. It is proventhat R is Artinian if and only if (GPC ,GIC) is a duality pair if and only if (GIC ,GPC)is a duality pair and M+ ∈ GIC whenever M ∈ GPC , where GPC (GIC) is the class ofC-Gorenstein projective (C-Gorenstein injective) R-modules. As applications, somenew criteria for a semidualizing module to be dualizing are given provided that R is acommutative Artinian ring. This talk is a report on joint work with Y. Geng and J.Hu.

Keywords: duality pairs; C-Gorenstein projective modules; semidualizing modules;dualizing modules

2010 Mathematics Subject Classification: 18G10; 18G15; 18G20.



2.29 Yilmaz DURGUN

Neat-Flat Modules

Yilmaz DURGUN, Engin BUYUKASIK

Izmir Institute of Technology, Department of Mathematics, Gulbahce Koyu, 35430,Urla, Izmir, Turkey.


Abstract: Throughout, R is an associative ring with identity and all modules areunitary right R-modules. For an R-module M , M+, E(M), Soc(M) will denote thecharacter module, injective hull, the socle of M , respectively. A subgroup A of anabelian group B is called neat in B if pA = A∩pB for each prime integer p. The notionof neat subgroup extended to modules by Renault (see, [6]). Namely, a submodule N ofR-module M is called neat in M , if for every simple R-module S, every homomorphismf : S → M/N can be lifted to a homomorphism g : S → M . Equivalently, N is neatin M if and only if Hom(S, g) : Hom(S,M) → Hom(S,M/N) is an epimorphism forevery simple R-module S. An R-module M is called m-injective if for any maximalright ideal I of R, any homomorphism f : I → M can be extended to a homomorphismg : R → M (see, [2], [3], [8], [9], [10]).

It turns out that, a module M is m-injective if and only if Ext1R(R/I,M) = 0 forany maximal right ideal I of R if and only if M is a neat submodule in every modulecontaining it i.e. any short exact sequence of the form 0 → M → N → L → 0 is neat-exact (see, [2,Theorem 2]). A ring R is a right C-ring if for every proper essential rightideal I of R, the module R/I has a simple module, (see, [7]). Any right semiartinianring is a C-ring, and a domain is a C-ring if and only if every torsion R-module containsa simple module. By [8,Lemma 4], R is a right C-ring if and only if every m-injectivemodule is injective.

Motivated by the relation between m-injective modules and neat submodules, we inves-tigate the modules M , for which any short exact sequence ending with M is neat-exact.Namely, we call M neat-flat if for any epimorphism f : N → M , the induced mapHom(S, N) → Hom(S,M) is surjective for any simple right R-module S.

In [4], a right R-module M is called simple-projective if for any simple right R-moduleN , every homomorphism f : N → M factors through a finitely generated free rightR-module F , that is, there exist homomorphisms g : N → F and h : F → M such thatf = hg. Simple-projective modules and a generalization of these modules have beenstudied in [4] and [5], respectively. By using simple-projective modules, the authors,characterize the rings whose simple (resp. finitely generated) right modules haveprojective (pre)envelope. Clearly, projective modules and modules with Soc(M) = 0are simple-projective. Also, a simple right R-module is simple-projective if and onlyif it is projective.

We prove the following results.

Theorem. A right R-module M is simple-projective if and only if M is neat-flat.

Proposition. The following are equivalent for a right R-module M :



(1) Soc(RR) = 0.

(2) M is neat-flat right R-module if and only if Soc(M) = 0.

Theorem. The following are equivalent for a ring R.

(1) Every neat-flat right R-module is projective.

(2) R is a right∑

-CS ring.

Theorem. Let R be a ring. The following are equivalent.

(1) Every finitely generated neat-flat right R-module is projective.

(2) R is a right C-ring and every finitely generated free right R-module is extending.

Theorem.The following conditions are equivalent for any ring R.

(1) R is a right Kasch ring.

(2) Every injective right R-module is neat-flat.

(3) The injective hull of every simple right R-module is neat-flat.

Keywords: closed submodule, neat submodule, extending modules, m-injective mod-ule, C-ring, CS-ring


Reference:

[1] L. Mao, When does every simple module have a projective envelope?, Comm.Algebra 35 (2007), 1505–1516.

[2] R. Parra and J. Rada, Projective envelopes of finitely generated modules, AlgebraColloq. 18 (2011), 801–806.

[3] G. Renault, Etude des sous-modules complements dans un module. Bull. Soc.Math. France Mem. (1967)(9).

[4] G. Renault, Etude de certains anneaux A lies aux sous-modules complements duna-module. C. R. Acad. Sci. Paris 259 (1964), 4203–4205.

[5] P. F. Smith, Injective modules and prime ideals, Comm. Algebra 9(9) (1981),989–999.

[6] M. Y. Wang and G. Zhao, On maximal injectivity, Acta Math. Sin. (Engl. Ser.)21(6) (2005), 1451–1458.

Y. Xiang, Max-injective, max- at modules and max-coherent rings, Bull. KoreanMath. Soc. 47(3) (2010), 611–622.



2.30 Ozgur EGE

On the Nilpotence of Steenrod Squares

Ozgur EGE1, Ismet KARACA2

1 Department of Mathematics, Celal Bayar University, Muradiye, Manisa, [email protected]

2 Department of Mathematics, Ege University, Bornova, Izmir, [email protected]

Abstract: In this paper, we deal with nilpotency of Steenrod operations. The de-termination of nilpotence height of an element the Steenrod Algebra A is a problemwhich has been researched since 1952. We state the structure of the Steenrod Algebrawith its main properties. At the end of the work, we give a conjecture about thenilpotence height of Steenrod powers.

Keywords: Steenrod Algebra, Steenrod squares, Nilpotency

2010 Mathematics Subject Classification: 55S05;55S10;57T05

Reference:

[1] Adem, J. The iteration of Steenrod squares in algebraic topology. Proc Nat AcadSci USA 38 (1952), 720–726.

[2] Adem, J. The relations in Steenrod powers of cohomology classes. Algebraic geom-etry and topology, Symposium in honour of S. Lefschetz: Princeton UniversityPress (1957).

[3] Karaca, I. The nilpotence height of P st for odd primes. Trans. Amer. Math. Soc.

351 (1999), 547–558.

[4] Karaca, I. Nilpotence Relations in the mod p Steenrod Algebra. Journal of Pureand Applied Algebra 171 (2002), 257–264.

[5] Karaca, I. Some formulas for nilpotence of Steenrod operations. InternationalJournal of Mathematics and Analysis, 1 (2006), no. 1, 69–84.

[6] Milnor, J. The Steenrod Algebra and its dual. Annals of Math. 67 (1958), 150–171.

[7] Monks, K.G. Nilpotence in the Steenrod algebra. Bol. Soc. Mat. Mexicana 37(1992), 401–416.

[8] Monks, K.G. The Nilpotence Height of P st . Proc. Amer. Math. Soc. 124 (1994),

1296–1303.

[9] Serre, J.P. Cohomologie module 2 des complexes d’Eilenberg-Mac-Lane. CommMath Helv 27 (1953), 198–232.

[10] Steenrod, N.E., Epstein, D.B.A. Cohomology operations. Annals of Math Studies50: Princeton University Press (1962).

[11] Walker, G., Wood, R.M.W. The nilpotence height of Sq2n

. Proc. Amer. Math.Soc. 124 (1996), 1291–1295.

[12] Walker, G., Wood, R.M.W. The nilpotence Height of P pn . Math. Proc. Camb.Phil. Soc. 123 (1998), 85–93.



2.31 Temha ERKOC

Rational Transitive Subgroups of Symmetric Groups

Temha ERKOC

Department of Mathematics, University of Istanbul, Istanbul, [email protected]

Abstract: A finite group G all of whose complex character values are rational num-bers is called a rational group. Equivalently, G is a rational group if and only ifNG(〈x〉)/CG(〈x〉) ∼= Aut(〈x〉) for every x ∈ G. All symmetric groups and their Sylow2-subgroups are rational groups. More generally, the Weyl groups of types An, Bn

and Dn,as well as their Sylow 2-subgroups, are all rational groups. In this talk, wedetermine rational transitive subgroups of some special symmetric groups. we alsoobtain several results releated to the study of rational groups.

Keywords: Rational groups, symmetric groups


Reference:

[1] Cameron P.J. Finite Permutation Groups and Finite Simple Groups, Bull. LondonMath. Soc., 13 (1981), 1 - 22

[2] Csorley, J.P. Cyclic Permutations in Doubly-Transitive Groups, Comm. Algebra,25 (1997), no.1, 33-35

[3] Feit W.-Seitz G.M.:On Finite Rational Groups and Related Topics, Illinois Journalof Mathematics, 33 (1988),103-131

[4] Kletzing D. Structure and Representations of Q-Groups, Springer-Verlag NewYork Berlin Heidelberg, (1984), 3-540-13865-X

[5] Gow R. Groups whose characters are rational-valued, Journal of Algebra, 40(1976), 280-299

[6] Chillag D. On a problem of Wielandt concerning permutation groups of primedegree, J. Algebra, 46 (1977), no. 1, 290-291

[7] Wielandt H. Finite Permutation Groups, Academic Press, New York/London,(1964)



2.32 Sehmus FINDIK

Weitzenbock Derivations of Free Metabelian AssociativeAlgebras

Rumen DANGOVSKI1, Vesselin DRENSKY2 and Sehmus FINDIK3

1 Sofia High School of Mathematics, 61, Iskar Str., 1000 Sofia, [email protected]

2 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113Sofia, Bulgaria

[email protected]

3 Department of Mathematics, Cukurova University, 01330 Balcalı, Adana, [email protected]

Abstract: A nonzero locally nilpotent linear derivation δ of the polynomial algebraK[Xd] = K[x1, . . . , xd] in several variables over a field K of characteristic 0 is calleda Weitzenbock derivation. The classical theorem of Weitzenbock [12] states that thealgebra of constants K[Xd]δ (which coincides with the algebra of invariants of a singleunipotent transformation) is finitely generated. Modern proofs of this theorem can befound in [10] and [11]. It is also known, see [7], [9], and [1], that one can define anaction of the special linear group SL2(K) on K[Xd+2] such that the algebra K[Xd]δ

and the algebra of invariants K[Xd+2]SL2(K) are isomorphic.

Similarly one may consider the algebra of constants Fd(V)δ of a locally nilpotent linearderivation δ acting on a finitely generated (not necessarily commutative or associative)algebra which is relatively free in a variety V of algebras over K. Now the algebra ofconstants is usually not finitely generated. In the case of associative algebras there isa dichotomy. If the variety of algebras V satisfies an identity which does not hold forthe algebra U2(K) of 2×2 upper triangular matrices, then Fd(V)δ is finitely generated[4]. Otherwise, if δ 6= 0, then Fd(V)δ is not finitely generated [5]. From this point ofview the free associative metabelian algebra Fd = Fd(N2A)δ, which is the relativelyfree algebra of the variety N2A generated by U2(K) (and defined by the polynomialidentity [x1, x2][x3, x4] = 0) is crucial for the investigation.

We show that the vector space of the constants (F ′d)δ in the commutator ideal F ′

d is afinitely generated K[X2d]δ-module. For small d, we calculate the Hilbert series of (F ′

d)δ

and find the generators of the K[X2d]δ-module (F ′d)δ. This gives also an (infinite) set

of generators of the algebra F δd . To calculate the Hilbert series we use a version of the

Elliot-McMahon method [6,8] following the ideas of [2] and [13].

This work may be considered as a continuation of our work [3] where we solved similarproblems for the free metabelian Lie algebra.

Keywords: Free metabelian associative algebras; algebras of constants; Weitzenbockderivations

2010 Mathematics Subject Classification: 16A06; 17B30; 17B40; 13N15; 13A50



Reference:

[1] Bedratyuk, L. Weitzenbock derivations and classical invariant theory: I. Poincareseries. Serdica Math. J. 36 (2010), 99–120.

[2] Benanti, F., Boumova, S., Drensky, V., Genov, G. K., Koev, P. Computing with ra-tional symmetric functions and applications to invariant theory and PI-algebras.Serdica Math. J. 38 (2012), 137–188.

[3] Dangovski, R., Drensky, V., Fındık, S. Weitzenbock derivations of free metabelianLie algebras. Linear Algebra and its Applications, to appear.

[4] Drensky, V. Invariants of unipotent transformations acting on noetherian relativelyfree algebras. Serdica Math. J. 30 (2004), 395–404.

[5] Drensky, V., Gupta, C. K. Constants of Weitzenbock derivations and invariantsof unipotent transformations acting on relatively free algebras. J. Algebra 292(2005), 393–428.

[6] Elliott, E. B. On linear homogeneous diophantine equations. Quart. J. Pure Appl.Math. 34 (1903), 348–377.

[7] Grosshans, F. Observable groups and Hilberts fourteenth problem. Amer. J.Math. 95 (1973), 229–253.

[8] MacMahon, P. A. Combinatory Analysis, vols. 1 and 2, Cambridge Univ. Press.1915, 1916. Reprinted in one volume: Chelsea, New York, 1960.

[9] Pommerening, K. Invariants of unipotent groups A survey. In: Invariant theory.Symp. West Chester/Pa. (1985), Lect. Notes Math. vol. 1278 (1987), 8–17.

[10] Seshadri, C. S. On a theorem of Weitzenbock in invariant theory. J. Math. KyotoUniv. 1 (1962), 403–409.

[11] Tyc, A. An elementary proof of the Weitzenbock theorem. Colloq. Math. 78(1998), 123–132.

[12] Weitzenbock, R. Uber die Invarianten von linearen Gruppen. Acta Math. 58(1932), 231–293.

[13] Xin, G. A fast algorithm for MacMahon’s partition analysis. Electron. J. Comb.11 (2004), no. 1, Research paper R58.



2.33 Silvia Claudia GAVITO

Main Modules and Some Characterizations of Rings withGlobal Conditions on Preradicals

Francisco RAGGI1, Jose RIOS1, Hugo RINCON2,Rogelio FERNANDEZ-ALONSO3 and Silvia GAVITO3

1 Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, MexicoD.F., Mexico

[email protected]

2Facultad de Ciencias, Universidad Nacional Autonoma de Mexico, Mexico D.F.,Mexico

[email protected]

3 Universidad Autonoma Metropolitana-Iztapalapa, Mexico D.F., [email protected] , [email protected]

Abstract: Main injective modules, which determine every left exact preradical, wereintroduced in [6]. In this paper we consider those modules which determine everypreradical and we call them main modules. We prove that a main module exists if,and only if, the lattice of preradicals R-pr is a set. Some properties of main modules areproven. We also prove some characterizations of rings for which (a) every preradicalis left exact, (b) every preradical is idempotent, (c) every preradical is a radical, (d)every preradical is a t-radical, (e) every preradical which is not the identity functoris prime. These characterizations relate to semisimple artinian rings, rings that area direct product of a finite number of simple rings, left V -rings, simple rings, amongothers.

Keywords: Main modules, preradicals, left exact preradicals, idempotent preradicals,radicals, t-radicals, prime preradicals, semisimple artinian rings, left V -rings, simplerings.

2010 Mathematics Subject Classification: 16S99; 16K99; 16S90

Reference:

[1] Fernandez-Alonso, R., Raggi, F., Rıos, J., Rincon, H., Signoret, C. The latticestructure of preradicals. Comm. Algebra 30 (2002), no. 3, 1533–1544.

[2] Fernandez-Alonso, R., Raggi, F., Rıos, J., Rincon, H., Signoret, C. The latticestructure of preradicals II. Partitions. J. Algebra Appl. 1 (2002), no. 2, 201–214.

[3] Fernandez-Alonso, R., Raggi, F., Rıos, J., Rincon, H., Signoret, C. The latticestructure of preradicals III. Operators. J. Pure Appl. Algebra 190 (2004), 251–265.



[4] Fernandez-Alonso, R., Raggi, F., Rıos, J., Rincon, H., Signoret, C. Prime andIrreducible Preradicals. J. Algebra Appl. 4 (2005), no. 4, 451–466.

[5] Fernandez-Alonso, R., Raggi, F., Rıos, J., Rincon, H., Signoret, C. Semiprimepreradicals. Comm. Algebra 37 (2009), 2811–2822.

[6] Fernandez-Alonso, R., Raggi, F., Rıos, J., Rincon, H. Basic Preradicals and MainInjective Modules. J. Algebra Appl. 8 (2009), no. 1, 1–16.

[7] Fernandez-Alonso, R., Gavito, S., Raggi, F., Rıos, J., Rincon, H. SemicoprimePreradicals. J. Algebra Appl. 11 (2012), no. 6, 1250115 (12 pages).

[8] Raggi, R., Rıos, J., Wisbauer, R. Coprime preradicals and modules. J. Pure Appl.Algebra 200 (2005), 51–69.



2.34 Hanieh GOLMAKANI

Some results of 4⊗−Engel groups

Hanieh GOLMAKANI1, Elaheh MOHAMMADZADEH2

1 Department Of Mathematics, Faculty Of Science, Mashhad Branch, Islamic AzadUniversity, Mashhad, 91735-413, Iran

[email protected]

2 Department of Mathematics, Faculty of Sciences, Payame Noor University,19395-4697 Tehran, Iran.

[email protected]

Abstract: In 2003, Biddle and Kappe considered tensor analogues of Engel elementsof a group. Following them, P. Moravec (2005) presented some results on tensoranalogues of 2-Engel group. In this paper first we define a tensor nilpotent group,then we get some new result for the tensor analogues of 4-Engel groups. We provethat if G is a 4⊗-Engle group, x, y ∈ G and a = yx then < a, ay > is tensor nilpotentof class 2.

Reference:

[1] D.P. Biddle, L.C. Kappe R.F. Morse, On subgroups related to the tensor center.Glasg. Math. J. 42 (2003). no 2, 323-332.

[2] R. Brown, J.-L. Loday, Van Kampen theorems for diagrams of spaces. Topology26 (1987), no. 3, 311-335

[3] G. Ellis , Tensor products and q-crossed modouls, J. London Math. Soc. (2) 51(1995), 243-258.

[4] A. McDermott, The non abelian tensor product of group: computations and struc-tural results, Department of Mathematics facuality of arts national universityof ireland galway. 1998.

[5] Shaban Sedghi, avalable at http://www.math.kth.se/4ecm/abstracts/3.6.pdf



2.35 Mouloud GOUBI

An Important Result About the Mertens Function

Mouloud GOUBI

Department of Mathematics, UMMTO University, Tizi ouzou, [email protected]

Abstract: In this paper, we formulate the functional expression of g (x) =∑

n≤x

µ(n)n

and the Mertens function M (x) =∑

n≤x µ(n) associated to Mobius function in theHilbert spaceH = L2

([0,∞[; dt

t2

). The calculus go alone to give a good theoretical upper bound of

M (x).

Reference:

[1] H. Cohen, F. Dress and M. El Marraki, Explicit estimates for summatory functionslinked to the Mobius µ-function, Functiones et Approximatio, XXXVII.1, (2007),

[2] M. Deleglise and J. Rivat, Computing the summation of the Mobius function Ex-perimental Mathematics, Vol. 5, (1996), no. 4, 291–295.

[3] F. Dress, Fonction sommatoire de la fonction de Mobius 1. Majorations experimentales,Experimental Mathematics, Vol. 2, (1993), no. 2, 89–98.

[4] M. El Marraki, Fonction sommatoire de la fonctionde Mobius, 3, J. de Theoriedes Nombres de Bordaux. 7 (1995), 407–433.

[5] M. Goubi, A. Bayad, Proof of the Mobius conjecture revisited, Proc. JangjeonMath. Soc. 16 (2013), no. 2, 237–243.



2.36 Nico J. GROENEWALD

Properties of different prime radicals of monoid modules

Nico J. GROENEWALD, David SSEVVIIRI

Department of Mathematics and Applied Mathematics,Nelson Mandela MetropolitanUniversity


Abstract: We investigate properties of different monoid module radicals arising fromthe different definitions of “prime” modules. Let R be a unital ring, M an R-moduleand G a monoid. If γ is a prime (resp. strongly prime and completely prime) radical ofa monoid module M(G), then γ(M [G]) = γ(M)[G]; (γ(M [G])∩M)[G] = γ(M [G]),i.e.,γ satisfies the Amitsur property; and if γ(M) = M , then γ(M [G]) = M [G], i.e.,γ ispolynomially extensible if M(G) = M [x]. We also show that a module RM is 2-primalif and only if the monoid module R(G)M(G) is 2-prima

Keywords: Unique product monoid, Radicals of modules, Amitsur property, 2-primalmodules

2010 Mathematics Subject Classification: 13C05, 16N40, 16N60, 16N80, 16S90



2.37 Eylem GUZEL KARPUZ

Grobner-Shirshov Bases and Word Problem for SomeGroups

Eylem GUZEL KARPUZ

Department of Mathematics, Karamanoglu Mehmetbey University, Karaman, [email protected]

Abstract: The Grobner-Shirshov basis theory was developed by A. I. Shirshov for Liealgebras [5] and B. Buchberger for commutative algebras [2]. It was also generalizedby G. M. Bergman and L. A. Bokut to the case of associative algebras [1]. Thistheory is very useful in the study of presentations of associative algebras, Lie algebras,semigroups and groups by generators and defining relations. It is also a powerful toolto solve many problems; normal form, word problem, embedding theorems, etc.

In this talk, after giving some important results on Grobner-Shirshov basis theory andword problem, I will present Grobner-Shirshov bases of some groups, namely extendedmodular, extended Hecke, Picard groups [3] and some Weyl groups [4].

Keywords: Grobner-Shirshov basis, word problem, Weyl group

2010 Mathematics Subject Classification: 13P10; 16S15; 20F55

Reference:

[1] Bokut, L. A. Imbedding into simple associative algebras. Algebra Logic 15 (1976),117-142.

[2] Buchberger, B. An algorithm for finding a basis for the residue class ring of azero-dimensional ideal. Ph.D. Thesis, University of Innsbruck, 1965.

[3] Karpuz, E. G., Cevik, A. S. Grobner-Shirshov bases for extended modular, ex-tended Hecke, and Picard groups. Mathematical Notes 92 (5) (2012), 636-642.

[4] Karpuz, E. G., Ates, F., Cevik, A. S. Grobner-Shirshov bases of some Weyl groups.Rocky Mountain J. Math. (accepted).

[5] Shirshov, A. I. Some algorithmic problems for Lie algebras. Siberian Math. J. 3(1962), 292-296.



2.38 Serpil GUNGOR

Co-coatomically Weak Supplemented Modules

Rafail ALIZADE1, Serpil GUNGOR2

1 Department of Mathematics, Yasar University, Bornova, Izmir, [email protected]

2 Department of Mathematics, Izmir Institute of Technology, Urla, Izmir, [email protected]

Abstract: In this paper it is shown that every supplement in a co-coatomicallyweak supplemented module M (i.e., every submodule N of M with M/N coatomic,has a weak supplement) is co-coatomically weak supplemented. If a module M isco-coatomically weak supplemented then every finitely M -generated module is co-coatomically weak supplemented module. Every left R-module is co-coatomically weaksupplemented if and only if the ring R is left perfect. Over a Dedekind domain, if thetorsion part T (M) of M has a weak supplement in M then M is co-coatomically weaksupplemented if and only if T (M) and M/T (M) are co-coatomically weak supple-mented.

Keywords: Coatomic module, supplement submodule, co-coatomic submodule, co-coatomically weak supplemented module.


Reference:

[1] Clark, J., Lomp, C., Vanaja, N., Wisbauer, R. Lifting Modules: Supplements andProjectivity in Module Theory. Basel:Birkhauser Verlag.

[2] Alizade, R., Buyukasık, E. Cofinitely Weak Supplemented Modules. Communica-tions In Algebra. 31 (2003), no. 11, 5377–5390.



2.39 Orhan GURGUN

Strong J-cleanness of Formal Matrix Rings

Orhan GURGUN1, Sait HALICIOGLU1 ans Abdullah HARMANCI2

1 Department of Mathematics, Ankara University, Tandogan, Ankara, [email protected] and [email protected]

2 Department of Mathematics, Hacettepe University, Beytepe, Ankara, [email protected]

Abstract: An element a of a ring R is called strongly J-clean provided that thereexists an idempotent e ∈ R such that a− e ∈ J(R) and ae = ea. A ring R is stronglyJ-clean in case every element in R is strongly J-clean. In this paper, we investigatestrong J-cleanness of M2(R; s) for a local ring R and s ∈ R. We determine theconditions under which elements of M2(R; s) are strongly J-clean.

Keywords: Strongly J-clean, strongly clean, strongly nil clean, formal matrix ring,local ring.

2010 Mathematics Subject Classification: 16S50; 16S70; 16U99

Reference:

[1] Chen, H. Rings Related Stable Range Conditions. Series in Algebra 11, WorldScientific, Hackensack, NJ, 2011.

[2] Chen, H. On strongly J-clean rings. Comm. Algebra 38 (2010), 3790-3804.

[3] Chen, H. Strongly J-clean matrices over local rings. Comm. Algebra (4) 40 (2012),1352-1362.

[4] Diesl, A. J. Classes of Strongly Clean Rings. Ph.D. Thesis, University of California,Berkeley, 2006.

[5] Krylov, P.A. Isomorphism of generalized matrix rings. Algebra Logic (4) 47 (2008),258-262.

[6] Krylov, P.A., Tuganbaev, A.A. Modules over formal matrix rings. J. Math. Sci.(2) 171 (2010), 248-295.

[7] Lam, T.Y. A First Course in Noncommutative Rings. Graduate Texts in Mathe-matics, vol. 131, Springer-Verlag, New York, 1991.

[8] Nicholson, W.K. Lifting idempotents and exchange rings. Trans. Amer. Math.Soc. 229 (1977), 269–278.

[9] Tang, G., Li, C., Zhou, Y. Study of Morita contexts. Comm. Algebra, in press.

[10] Tang, G., Zhou, Y. Strong cleanness of generalized matrix rings over a local ring.Linear Algebra Appl. (10) 437 (2012), 2546-2559.

[11] Tang, G., Zhou, Y. A class of formal matrix rings. Linear Algebra Appl. (12)438 (2013), 4672-4688.

[12] Yang, X., Zhou, Y. Strongly cleanness of the 2 × 2 matrix ring over a generallocal ring. J. Algebra 320 (2008), 2280-2290.



2.40 Claus HAETINGER

A note on Homomorphisms and Anti-homomorphisms on∗-ring

Claus Haetinger1, Nadeem UR REHMAN2 and Abu Zaid ANSARI2

1 Center of Exact and Technological Sciences, UNIVATES University Center,Lajeado-RS, [email protected]

2 Department of Mathematics, Aligarh Muslim University, Aligarh, [email protected] and [email protected]

Abstract: In this paper we describe generalized left ∗-derivation F : R → R in ∗-prime ring and prove that if F acts as homomorphism or anti-homomorphism on R,then either R is commutative or F is a right ∗-centralizer on R. Analogous resultshave been proved for generalized left ∗-biderivation and Jordan ∗-centralizer on R

Partially supported by Coordenaccao de Aperfeicoamento de Pessoal de nıvel Superior- CAPES (Brazil).

Keywords: Semiprime (prime) ring; involution; generalized left ∗-derivation; gener-alized left ∗-biderivation and Jordan ∗-centralizer

Reference:

[1] Ali, S. On generalized left derivations in rings and Banach algebras, Aequ. Math.81 (2011), 209-226.

[2] Ashraf, M., Ali, S. On generalized Jordan left derivations in rings, Bull. KoreanMath. Soc. 45, No. 2, (2008), 253-261.

[3] Ashraf, M., Rehman, N. On Lie ideals and Jordan left derivations of prime rings,Arch. Math. (Brno) 36, (2000), 201-206.

[4] Bell, H.E., Kappe, L.E. Ring in which derivations satisfying certain algebraicconditions, Acta Math. Hungar. 53 (1989), 339-340.

[5] Bresar, M. Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218-228.

[6] Bressar, M. On generalized biderivations and related maps. J. Algebra 172, (1995),764-786.

[7] Bresar, M., Vukman, J. On left derivations and related mappings, Proc. Amer.Math. Soc. 110 (1990), 7-16.

[8] Bresar, M., Vukman, J. On some additive mappings in rings with involution, 38,(1989), 178-185.

[9] Herstein, I.N. Topics in ring theory, University of Chicago Press, Chicago, London,1969.

[10] Maska, G.Y. Remark on symmetric biadditive functions having non-negativediagonalization, Glasnik Matematicki 15, (1980), 279-280.



[11] Muthana, N.M. Left centralizer traces, generalized biderivations, left bi-multipliersand generalized Jordan biderivations, Aligarh Bull. Math. 26(2)(2007), 33-45.

[12] Rehman, N., Ansari, A.Z. Additive mappings of semiprime rings with involution,Alig. Bull. Math. 30(1-2) (2011), 1-7.

[13] Vukman, J. Two results concerning symmetric biderivations on prime rings, Ae-quationes Math. 40, (1990), 181-189.

[14] Zalar, B. On centralizers of semiprime rings, Comment. Math. Univ. carolin.32, 4 (1991), 609-614.



2.41 Khalid Ali Mohammad HAMDIN

Generalized (α, β)-derivations of σ-prime rings

Khalid Ali Mohammad HAMDIN

Aligarh Muslim University, Aligarh 202002 UP [email protected]

Abstract:



2.42 Ayazul HASAN

On α-Modules

Ayazul Hasan1, Alveera Mehdi2

1Department of Mathematics, Integral University, Lucknow, 226026, [email protected]

2Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, Indiaalveera [email protected]

Abstract: A right module M over an associative ring with unity is a QTAG-module ifevery finitely generated submodule of any homomorphic image of M is a direct sum ofuniserial modules. Let α denotes the class of QTAG-modules M such that M/Hβ(M)is totally projective, for all ordinals β < α. Naji defined these modules and called themα-modules. Here we study these modules in the light of nice submodules and obtainsome interesting results.

Keywords: QTAG-modules, nice submodules

2010 Mathematics Subject Classification: 16K20



2.43 Can HATIPOGLU

Injective Hulls of Simple Modules over Nilpotent LieSuperalgebras

Can HATIPOGLU, Christian LOMP

Department of Mathematics, University of Porto, [email protected] and [email protected]

Abstract: We consider Noetherian rings over which injective hulls of simple mod-ules are locally Artinian. Namely we show that the finite dimensional nilpotent Liesuperalgebras g whose injective hulls of simple U(g)-modules are locally Artinian areprecisely those whose even part g0 is isomorphic to a nilpotent Lie algebra with anabelian ideal of codimension 1 or to a direct product of an abelian Lie algebra and acertain 5-dimensional or a certain 6-dimensional nilpotent Lie algebra.

Keywords: Injective modules, Simple modules, Nilpotent Lie superalgebras, Maximalindex of Lie algebras

2010 Mathematics Subject Classification: 17B30;17B10



2.44 Ilker INAM

On the Sato-Tate Conjecture

Ilker INAM

Department of Mathematics, Uludag University, Gorukle, Bursa, [email protected] and [email protected]

Abstract: Let f :=∑∞

n=1 a(n)qn be a normalised cuspidal Hecke eigenform of weight

2k for Γ0(N) without complex multiplication. Define b(p) := a(p)

2pk−1/2 for prime p.

Then the Sato-Tate conjecture, which is a theorem now after Barnet-Lamb, Geraghty,Harris and Taylor, says that the numbers b(p) are equidistributed in [−1, 1] withrespect to ”the Sato-Tate measure”, when p runs through the primes not dividing N .This theorem is one of the most important results in number theory recently.

In this talk, we will introduce the Sato-Tate conjecture and explain how one can useit in half-integral weight modular forms using the Shimura lift.

Keywords: Modular forms, Half-integral weight modular forms, equidistribution

2010 Mathematics Subject Classification: 11F30;11F37;11F11

Reference:

[1] Arias-de-Reyna, S., Inam, I., Wiese, G. On Conjectures of Sato-Tate and Bruinier-Kohnen, preprint, (2013),

[2] Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R. A Family of Calabi-Yauvarities and Potential Automorphy II, Publications of the Research Institute forMathematical Sciences, 47 (2011), no. 1, 29–98,

[3] Bruinier, J.H., Kohnen, W. Sign Changes of Coefficients of Half Integral WeightModular Forms, Modular forms on Schiermonnikoog, Eds.: B. Edixhoven, G.van der Geer and B. Moonen, Cambridge University Press (2008), 57–66,

[4] Inam, I., Wiese, G. Equidistribution of Signs for Modular Eigenforms of HalfIntegral Weight, preprint, (2013),

[5] Kohnen, W. A Short Note on Fourier Coefficients of Half-Integral Weight ModularForms, Int. J. of Number Theory, 06 (2010), 1255–1259,

[6] Kohnen, W., Lau, Y.-K., Wu, J. Fourier Coefficients of Cusp Forms of Half-IntegralWeight, Math. Zeitschrift, 273 (2013), 29–41,

[7] Kohnen, W., Zagier, D. Values of L−series of Modular Forms at the Center of theCritical Strip, Invent. Math., 64 (1981), 175–198,

[8] Mazur, B. Finding Meaning in Error Terms, Bulletin of the Amer. Math. Soc.,45 no:2 (2008), 185–228,

[9] Shimura, G. On Modular Forms of Half-Integral Weight, Annals of Math., 97(1973) 440–481.



2.45 Berke KALEBOGAZ

Direct Sums of Semi-projective Modules

Derya KESKIN TUTUNCU1, Berke KALEBOGGAZ1 and Patrick F. SMITH2

1 Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, [email protected] and [email protected]

2 Department of Mathematics, Glasgow University, Glasgow G12 8QW, [email protected]

Abstract: Let R be a ring. An R-module M is called semi-projective provided for allendomorphisms α and β of M with β(M) ⊆ α(M) there exists an endomorphism γ ofM such that β = αγ. In this work, we give some basic properties of semi-projectivemodules and provide some characterizations as follows:

Corollary: Every nonsingular extending module is semi-projective.

Theorem: Let R be any Dedekind domain. Then the following statements are equiv-alent for an R-module M which is a direct sum of cyclic submodules.

(i) M is quasi-projective.

(ii) M is semi-projective.

(iii) M is direct projective.

Also it is shown that every direct summand of a semi-projective module inherits theproperty, while a direct sum of semi-projective modules need not be semi-projective.And we get that;

Remark: Let a module M = ⊕i∈I Mi be a direct sum of submodules Mi (i ∈ I) suchthat HomR(Mi,Mj) = 0 for all i 6= j in I. Then M is semi-projective if and only ifMi is semi-projective for all i ∈ I.

Then we study direct sums of semi-projective modules over right Ore domains and weprove that:

Corollary: Let R be a right Ore domain with right quotient division ring Q 6= R andX is a free right R-module then the right R-module Q ⊕ X is semi-projective if andonly if there does not exist an R-epimorphism from X to Q.

We observe that;

Theorem: Let R be a PID with field of fractions Q and let X be a proper submoduleof Q such that R ⊆ X. Then the following statements are equivalent for the R-moduleM = X ⊕R.

(i) M is finitely generated.

(ii) M is projective.

(iii) M is semi-projective.



(iv) M is direct projective.

Keywords: Semi-projective module, direct sum, rigth Ore domain.

2010 Mathematics Subject Classification: 16D40;16U10;16D50;13C11;13F10

Reference:

[1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York (1974).

[2] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules Birkhauser Ver-lag, Basel (2006).

[3] L. Fuchs and K. M. Rangaswamy, Quasi-projective abelian groups, Bull. Soc.Math., France, 98 (1970), 5–8.

[4] K. R. Goodearl and R. B. Warfield, Jr., An Introduction to NoncommutativeNoetherian Rings, London Math. Soc. Student Texts 16, Cambridge Univ.Press, Cambridge (1989).

[5] A. Haghany and M. R. Vedadi, Study of semi-projective retractable modules,Algebra Coll. 14 (3) (2007), 489–496.

[6] S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Math.Soc. Lecture Series 147, Cambridge Univ. Press, Cambridge (1990).

[7] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, Chichester (1987).

[8] K. M. Rangaswamy, Abelian groups with endomorphic images of special types, J.Algebra 6 (1967), 271–280.

[9] H. Tansee and S. Wongwai, A note on semi-projective modules, Kyungpook Math.J. 42 (2002), 369–380.

[10] R. Wisbauer, Foundations of Module and Ring Theory Gordon and Breach,Philadelphia (1991).

[11] R. Wisbauer, Modules and Algebras: Bimodule Structure On Group Actions andAlgebras, Pitman Monographs 81, Longman (1996).



2.46 Tolga KARAYAYLA

On the Structure of Automorphism Groups of RationalElliptic Surfaces

Tolga KARAYAYLA

Department of Mathematics, Middle East Technical University, Ankara, [email protected]

Abstract: In this talk, I will present the two step project of the classification ofthe automorphism groups of rational elliptic surfaces (RES) with section over the basefield C. The first step of the project is concerned with those RES with non-constant J-map, and the second step is concerned with the RES with constant J-map. The groupAut(B) of the regular automorphisms (biholomorphic maps) of a relatively minimalRES B with section has the structure of a semi-direct product MW (B) ⋊ Autσ(B)where MW (B) is the group of sections on the surface B and Autσ(B) is the subgroupof automorphisms which preserve the zero section σ. The Mordell-Weil group MW (B)has been classified by Oguiso and Shioda with respect to the configurations of singularfibers on B. In this project, we determine the group Autσ(B) with respect to theconfigurations of singular fibers on B. The difference between the constant and non-constant J-maps is that the group Autσ(B) shows richer structure when the J-map isconstant. The group Autσ(B) has size at most 24 for non-constant J-map case whileit can have size 144 or can even be an infinite group in the constant J-map case. Thisdifference between the two cases is due to the facts that when the J-map is constantall smooth elliptic curve fibers are isomorphic, and there are automorphisms that actas complex multiplication of order 3, 4 or 6 (depending on the constant J value).

Keywords: Automorphism Group, Elliptic Surface, Rational Elliptic Surface, Mordell-Weil Group, Singular Fiber, J Map.

2010 Mathematics Subject Classification: 14J50; 14J27; 14J26



2.47 Abidin KAYA

Quadratic Residue Codes over Fp + vFp

Abidin KAYA1, Bahattin YILDIZ1 and Irfan SIAP2

1 Department of Mathematics, Fatih University, Istanbul, [email protected] and [email protected]

2 Department of Mathematics, Yıldız Technical University, Istanbul, [email protected]

Abstract: Quadratic residue (QR) codes are an interesting family of cyclic codes.The binary Golay code falls into this category. In recent years, quadratic residuecodes over some special rings such as Z4,Z8 and Z9 are studied in detail respectivelyin [3,1,4].

In this work, quadratic residue codes over the non-chain, non-local ring Fp+vFp wherev2 = v are introduced. They are defined in terms of their idempotent generators andit is observed that they share the properties of QR codes over fields. For the casep = 2 two Hermitian self-dual codes which are optimal with respect to the Bachocbound are obtained. Moreover, we had some extremal self-dual and optimal self-dualp-ary codes as Gray images of the extended QR codes over Fp +vFp where p is an oddprime. Some of these codes coincides with quadratic double circulant (QDC) codesconstructed in [5]. The details and results are available online at [2].

Keywords: codes over rings, quadratic residue codes

2010 Mathematics Subject Classification: 94B05,94B15

Reference:

[1] Chiu, M.H., Yau, S.T., Yu, Y., “Z8-cyclic codes and quadratic residue codes”,Adv. Appl. Math, Vol. 25, pp.12–33, 2000.

[2] Kaya, A. Yıldız, B. Siap, I., “Quadratic residue codes over Fp+vFp and their Grayimages” in submission, available online at http://arxiv.org/abs/1305.4508.

[3] Pless, V., Qian, Z., “Cyclic codes and quadratic residue codes over Z4”, IEEETrans. Inform. Theory, Vol.42, No.5, pp. 1594-1600, 1996.

[4] Taeri, B., “Quadratic Residue Codes over Z9”, J. Korean Math. Soc., Vol.46,No.1, pp. 13-30, 2009.

[5] Gaborit, P., “Quadratic double circulant codes over fields”, Vol. 97, Issue 1, pp.85-107, 2002.

[6] Zhu, S.X., Wang, L., “A class of constacyclic codes over Fp + vFp and its Grayimage”, Discrete Mathematics, Vol. 311, pp. 2677–2682, 2011.



2.48 Shahoor KHAN

Identities with Generalized Derivations andAutomorphisms in Rings

Shahoor KHAN

Department of Mathematics, Aligarh Muslim University, [email protected]

Abstract:



2.49 Sibel KILICARSLAN CANSU

On Generalized Semiradical Formula

Sibel KILICARSLAN CANSU1, Erol YILMAZ2

1 Department of Mathematics, Istanbul Technical University, Maslak, Istanbul,Turkey

[email protected]

2 Department of Mathematics, Abant Izzet Baysal University, Golkoy, Bolu, Turkeyyilmaz [email protected]

Abstract: A proper submodule N of an R-module M is called semiprime, if wheneverrkm ∈ N , then rm ∈ N for some r ∈ R,m ∈ M and k ∈ Z+. The semiradical of N ,sradM (N), is defined as the intersection of all semiprime submodules of M containingN . By using the semiradical of a submodule, the generalized semiradical formula isdefined and we showed that Noetherian modules satisfy the generalized semiradicalformula.

Keywords: Semiprime submodule, semiradical formula, envelope.

2010 Mathematics Subject Classification: 13C99;13A99

Reference:

[1] Azizi, A. Radical Formula and Prime Submodules. J. Algebra. 307 (2007), 454–460.

[2] Jenkins, J., Smith, P. F. On The Prime Radical of a Module Over A CommutativeRing. Comm. in Algebra. 20 (1992), no. 12, 3593–3602.

[3] McCasland, R. L., Moore, M. E. On Radicals of Submodules. Comm. in Algebra.19 (1991), no. 5, 1327–1341.

[4] Tavallaee, H. A., Varmazyar, R. Semi-Radicals of Submodules in Modules. IUSTInternational Journal of Engineering Science. 19 (2008), no. 1-2, 21–27.



2.50 M. Tamer KOSAN

On ADS Modules and Rings

Truong Cong QUYNH1, M. Tamer KOSAN2

1 Department of Mathematics, Danang University 459 Ton Duc Thang, DaNang city,Vietnam

[email protected] , [email protected]

2 Department of Mathematics, Gebze Institute of Technology Cayirova Campus,41400, Gebze- Kocaeli, Turkey

[email protected]

Abstract: I will work out in ADS (Absolute Direct Summand) modules. This isa class of modules between quasi-continuous modules and modules satisfying the C3condition. It was introduced by Fuchs [1] for abelian groups and for general modulesby Alahmani, Jain and Leroy [2]. In [3], we give different characterizations of ADSmodules and show how to characterize semisimple modules and semisimple artinianrings using the ADS. The SC and SI rings are also characterized by the ADS notion.

Keywords: ADS module, ADS ring, group module, group ring, fully-invariant mod-ule, CS module, C3 module

2010 Mathematics Subject Classification: 16D50;16D40

Reference:

[1] Fuchs, L. (1970): Infinite Abelian Groups. vol. I, Pure Appl. Math., Ser. Monogr.Textb., vol. 36, Academic Press, New York, San Francisco, London.

[2] Alahmadi, A., Jain, S.K., Leroy, A.: ADS modules. J. Algebra 352(2012), 215-222.

[3] Quynh , T.C., Kosan, M. T.: On ADS modules and rings. Commun. Algebra inpress.



2.51 Berna KOSAR

T-Generalized Supplemented Modules

Berna KOSAR, Celil NEBIYEV

Department of Mathematics, Ondokuz Mayıs University, Kurupelit, Samsun, [email protected] and [email protected]

Abstract: In this paper, t-generalized supplemented modules are defined by start-ing out from generalized ⊕-supplemented modules. Besides giving examples sepa-rating t-generalized supplemented modules, supplemented modules and generalized⊕-supplemented modules, we also show the equality of these for projective and finitelygenerated modules. Nevertheless we define cofinitely t-generalized supplemented mod-ules and give the characterization of these modules. Moreover, for any ring R, weinvestigate that any finite direct sum of t-generalized supplemented R−modules is at-generalized supplemented and arbitrary direct sum of cofinitely t-generalized supple-mented R−modules is a cofinitely t-generalized supplemented module.

Keywords: small submodules, radical, supplemented modules, generalized supple-mented modules

2010 Mathematics Subject Classification: 16D60, 16D80

Reference:

[1] Buyukasık, E., Lomp, C., On a recent generalization of semiperfect rings, Bulletinof the Australian Mathematical Society, 78(2008), no.2, 317-325.

[2] Clark, J., Lomp, C., Vanaja, N., Wisbauer, R., Lifting Modules. Supplements andProjectivity in Module Theory, Frontiers in Mathematics, Birkhauser, Basel(2006).

[3] Calısıcı, H., Turkmen, E., Generalized ⊕-supplemented modules, Algebra andDiscrete Mathematics, 10(2010), 10-18.

[4] Ecevit, S., Kosan, M.T., Tribak, R., Rad-⊕-supplemented modules and cofinitelyRad-supplemented modules, Algebra Colloq., 19(2012), 637-648.

[5] Idelhadj, A., Tribak, R., On some properties of ⊕-supplemented modules, Int. J.Math. Sci., 69(2003), 4373-4387.

[6] Kasch, F., Modules and Rings. Academic Press. London New-York, 1982.

[7] Mohamed, S.H., Muller, B.J., Continuous and discrete modules, London Math.Soc. LNS, 147, Cambridge Univ. Press, Cambridge, New-York, 1990.

[8] Talebi, Y., Hamzekolaei, A. R. M., Tutuncu, D. K., On rad ⊕-supplementedmodules, Hadronic Journal, 32(2009), 505-512.

[9] Talebi, Y., Mahmoudi, A., On rad ⊕-supplemented modules, Thai Journal ofMathematics, 9(2011), no.2, 373-381.



[10] Wang, Y., Ding, N., Generalized supplemented modules, Taiwanese Journal ofMathematics, 10(2006), no. 6, 1589-1601.

[11] Wisbauer, R., Foundations of Module and Ring Theory. Gordon and Breach,Philadelphia, 1991.

[12] Zoschinger, H., Komplementierte moduln uber dedekindringen, J.Algebra 29(1974),42-56.

[13] Xue, W., Characterizations of semiperfect and perfect rings, Publications Matem-atiques, 40(1996), 115-125.



2.52 Yosum KURTULMAZ

Strongly J-clean Skew Triangular Matrix Rings

Yosum KURTULMAZ

Department of Mathematics, Bilkent University, Ankara, [email protected]

Abstract: Let R be an arbitrary ring with identity. An element a ∈ R is stronglyJ-clean if there exist an idempotent e ∈ R and element w ∈ J(R) such that a = e+wand ew = ew. A ring R is strongly J-clean in case every element in R is strongly J-clean. In this note, we investigate the strong J-cleanness of the skew triangular matrixring Tn(R, σ) over a local ring R, where σ is an endomorphism of R and n = 2, 3, 4.

Keywords: strongly J-clean ring, skew triangular matrix ring, local ring.

2010 Mathematics Subject Classification: 16D70, 16E50.

Reference:

[1] Chen H. On strongly J-clean rings. Comm. Algebra 38(2010), 3790–3804.

[2] Chen H. On uniquely clean rings, Comm. Algebra 39(2011), 189–198.

[3] Diesel A.J.Classes of Strongly Clean Rings. Ph.D. Thesis, University of California,Berkeley, 2006.

[4] Li Y. On strongly clean matrix rings. J. Algebra 312(2007), 397–404.

[5] Nicholson W.K. and Zhou Y. Rings in which elements are uniquely the sum of anidempotent and a unit. Glasg. Math. J. 46(2004), 227–236.

[6] Nicholson W.K. Clean rings: a survey. Advances in Ring Theory. World Sci.Publ. Hackensack, NJ, (2005), 181-198.

[7] Yang X. and Zhou Y. Some families of strongly clean rings. Linear Algebra Appl.425 (2007), 119–129.

[8] Yang X. and Zhou Y. Strongly cleanness of the 2 × 2 matrix ring over a generallocal ring. J. Algebra 320(2008), 2280–2290.



2.53 Semra KUS

q-Bernoulli Matrices and Stirling Numbers

Semra KUS 1, Naim TUGLU2

1 Mucur Vocational High School, Ahi Evran University, Mucur, Kırsehir, [email protected]

2 Department of Mathematics, Gazi University, Teknikokullar, Ankara, [email protected]

Abstract: In this study we define q−Bernoulli matrices by using q−Bernoulli numbersand q−Bernoulli polynomials. Also we defined special q-matrices by using first andsecond kind of Stirling numbers. Then, we obtain the factorization of q−Bernoullimatrices using these matrices.

Keywords: q−Bernoulli numbers, q−Bernoulli matrices, Stirling numbers

2010 Mathematics Subject Classification: 11B68;11B73

Reference:

[1] Kac, V., Cheung P., “Quantum Calculus” Springer Newyork (2002)

[2] Lalin, M.,N., “Bernoulli Numbers” Junior Number Theory Seminar-Universty ofTexas at Austin September 6th (2005)

[3] Zhang, Z., Whang, J., “Bernoulli Matrix and its Algebraic Properties” DiscreteApplied Mathematics 154: 1622-1632 (2006)

[4] Ernst, T., “ q−Pascal and q−Bernoulli Matrices and Umbral Approach” Depart-ment of Mathematics Uppsala Universty U.U.D.M. Report 2008:23 (2008)

[5] Hegazi, A. S. and Mansour, M., “A Note on Bernoulli Numbers and Polynomials”Journual of Nonlinear Mathematical Physics 13(1): 9-18 (2005)

[6] Cheon, G.-S. and Kim, J.-S., “Factorial Stirling Matrix and Related Combinato-rialsequences” Linear Algebra and its Applications 357(1-3) 247-258 (2002)



2.54 Christian LOMP

Integral Calculus on Quantum Exterior Algebras

Serkan KARACUHA, Christian LOMP

Department of Mathematics, Faculty of Science, University of Porto, Porto, [email protected] and [email protected]

Abstract: Hom-connections and associated integral forms have been introduced andstudied by T.Brzezinski [1,2,3,4] as an adjoint version of the usual notion of a con-nection in non-commutative geometry. Given a flat hom-connection on a differentialcalculus (Ω, d) over an algebra A yields the integral complex which for various algebrashas been shown to be isomorphic to the de Rham complex. In this paper we shed fur-ther light on the question when the integral and the de Rham complex are isomorphicfor an algebra A with a flat hom-connection. We specialise our study to the case wherean n-dimensional differential calculus can be constructed on a quantum exterior alge-bra over an A-bimodule. Criteria are given for free bimodules with diagonal or uppertriangular bimodule structure. Our results are illustrated for a differential calculus ona multivariate quantum polynomial algebra and for a differential calculus on Manin’squantum n-space.

Keywords: Non-commutative geometry, differential graded algebra, Quantum Exte-rior Algebra, Hom-connections

2010 Mathematics Subject Classification: 58B32; 16W25

Reference:

[1] T. Brzezinski, Non-commutative connections of the second kind, J. Algebra Appl.7 (2008), 557573

[2] T. Brzezinski, Integral calculus on Eq(2), SIGMA Symmetry Integrability Geom.Methods Appl. 6 (2010), Paper 040, 10 pp.

[3] T. Brzezinski, Divergences on projective modules and non-commutative integrals.,Int. J. Geom. Methods Mod. Phys. 8(4) (2011), 885-896

[4] T. Brzezinski, L. El Kaoutit, C. Lomp, Non-commutative integral forms andtwisted multi-derivations, J. Noncommut. Geom. 4 (2010), 281312

[5] S.Karacuha, C. Lomp, Integral Calculus on Quantum Exterior Algebras, arXiv1302.5216v2 (2013)



2.55 Adolf MADER

Representations of Posets and Decompositions ofTorsion-free Abelian Groups

David ARNOLD1, Adolf MADER2, Otto MUTZBAUER3, Ebru SOLAK4

1 Department of Mathematics, Baylor University, Waco, Texas 76798-7328, USADavid [email protected]

2 Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu,Hawaii 96822, USA

[email protected]

3 Universitat Wurzburg, Mathematisches Institut, Am Hubland, 97074 Wurzburg,Germany

[email protected]

4 Department of Mathematics, Middle East Technical University, Inonu Bulvarı,06531 Ankara, [email protected]

Abstract: Torsion-free abelian groups are additive subgroups of rational vectorspaces.

An almost completely decomposable group is a torsion-free abelian group G containinga subgroup R that is a finite direct sum of additive subgroups of the rational field Qwith G/R finite.

Representations of posets in certain modules are used to discuss direct decomposi-tions of almost completely decomposable groups. For almost completely decomposablegroups with p-primary quotient group G/R direct decompositions are unique up tonear-isomorphism, a weakening of isomorphism. Among the categories of rigid almostcompletely decomposable groups G with G/R a direct sum of p-primary cyclic groupsall of the same order we determine those that contain indecomposable groups of anyfinite rank in which case a complete description is hopeless, and for the remaining caseswe completely determine the near-isomorphism classes of indecomposable groups. Forthe latter groups we have a analog to the basis theorem for finite abelian groups.

Keywords: poset, representation, almost completely decomposable, rigid, indecom-posable, near–isomorphism.

2010 Mathematics Subject Classification: 20K15; 20K25; 16G60

Reference:

[1] David M. Arnold, Abelian Groups and Representations of Partially Ordered Sets,CMS Advanced Books in Mathematics, Springer-Verlag, New York, 2000.

[2] Adolf Mader, Almost Completely Decomposable Groups, Algebra, Logic and Appli-cations Series Volume 13, Gordon and Breach, Amsterdam, 2000.



2.56 Phool MIYAN

On Semiderivations of PrimeNear Rings

Phool MIYAN

Department of Mathematics, Aligarh Muslim University, Aligarh, [email protected]

Abstract: As a generalization of derivations the notion of a semiderivation wasintroduced by Bergen [4]. An additive mapping f : N −→ N is said to be asemiderivation on a near ring N if there exists a function g : N −→ N suchthat (i) f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and (ii)f(g(x)) = g(f(x))hold for all x, y ∈ N . In case g is the identity map on N , f is of course justa derivation. Let N = N1 ⊕ N2, where N1 is a zero-symmetric near ring andN2 is a ring. Then the map f : N −→ N defined by f(x, y) = (0, y) is asemiderivation associated with function g : N −→ N such that g(x, y) = (x, 0). How-ever f is not a derivation on N . The purpose of the present paper is to extend some re-sults proved by Bell [1], Beidar, Fong and Wang [2] andBell and Mason [3] on derivation to semiderivation of a prime near ring.

Keywords: 3-prime near rings, derivations, semiderivations

2010 Mathematics Subject Classification: 16N60, 16W25, 16Y30

Reference:

[1] Bell, H. E. On derivations in near rings II, Kluwer Academic Publ. Math. Appl.Dordr. 426 (1997), 191-197.

[2] Beidar, K. I. , Fong, Y. and Wang, X. K. Posner and Herstein Theorems forderivations of 3-prime near rings, Comm. Algebra, 24 (1995), 1581-1589.

[3] Bell, H. E. and Mason, G. On derivations in near rings, Near rings and Near fields,North-Holland Mathematical Studies, 137 (1987), 31-35.

[4] Bergen, J. Derivations in prime rings, Canad. Math. Bull. 26 (1983),267-270.



2.57 Najat Mohammed MUTHANA

On Some Equations Related to (θ, θ)-derivations in Rings

Najat Mohammed MUTHANA

Department of Mathematics, King Abdulaziz University, Jeddah, Saudi [email protected]

Abstract: Let m and n be positive integers with m+n 6= 0 and let R be a (m+n+2)!−torsion free semiprime ring with the identity element suppose there exists an additivemapping D : R → R such that D(xm+n+1) = (m + n + 1)θ(xm)D(x)θ(xn) holds forall x ∈ R where θ is an epimorphisms of R. Then D is a (θ, θ)- derivation.

Keywords: Semiprime rings; derivation; nth-power property; (θ, θ)-derivation.

2010 Mathematics Subject Classification: 16N60; 39B05

Reference:

[1] M. Bresar and J. Vukman, Jordan derivations on prime rings, Bull Austral. Math.Soc 37(1988), 321-322.

[2] M. Bresar and J. Vukman, Jordan (θ, θ)-derivations, Glas. Math. Sec. III 26(46),(1991), no.(1-2), 13-17.

[3] M. Bresar and J. Vukman, On left derivations and related mappings, Proc. Amer.Math. Sec. 110, (1990), 7-16.

[4] D. Bridges and J. Bergen, On the derivation of xn in a ring, Proc. Amer. Math.Soc. 90(1984), no.1, 25-29.

[5] J.M. Cusak, Jordan derivations on rings, Proc. Amer. Math. Soc(53), 321-324.

[6] I.N. Herstein, Jordan derivation of prime rings, Proc. Amer. Math. Soc. 8(1957),1104-111.

[7] C. Lanski, Generalized derivations and nth power maps in rings, comm. Algebra,35, (2007), 3660-3672.

[8] N.Muthana and N. Daif, when is an additive mappings of a ring a (θ, θ)-or (θ, φ)-derivations. Global Journal of Pure and Applied Mathematics, Vol (4), No 2,(2008), 61-72.

[9] J. Vukman, commuting and centralizing mappings in prime rings, Proc. Amer.Math. Soc. 109, (1990), 47-52.

[10] J. Vukman, Jordan left derivations on semiprime rings. Math. J.J. OkayamaUniv, 39, (1997), 1-6.

[11] J. Vukman and I. Kosi-Ulbl, On Some equations related to derivations in rings,Int. J. Math. Math. Sci, 17, (2005), 2703-2710.



2.58 Sinem ODABASI

On Some Torsion Theories in the Category ofQuasi-coherent Sheaves

Sinem ODABASI

Departamento de Matematicas, Universidad de Murcia, Murcia, [email protected]

Abstract: The class of flat quasi–coherent sheaves on a scheme X has been exten-sively used during the last years, as a natural choice for studying both the homotopycategory and the derived category of quasi–coherent sheaves ([4,7]). It is known thatthe category (X) of quasi–coherent sheaves admits arbitrary arbitrary products. How-ever, it seems to be a hard task to know an explicit description of this object. This ispartially so because, at the level of sections, product of modules is not well-behavedin general with respect to localizations, or more generally, when tensoring by an ar-bitrary module with respect to a commutative ring (direct products do not commutewith tensoring in general). But even in case that the tensor product does commutewith products with respect to finitely presented modules (for instance when the ringR is coherent) it is not clear whether the product object in (X) can be computedfrom the product module of sections at each affine open if we do not impose extraassumptions on the sheaf of rings OX attained to X (for instance if OX(U) is finitelypresented as OX(V )-module, for each affine open subsets U ⊆ V ).

The lack of an explicit description of the product object leads to new and relevantquestions on the class Flat(X) of flat quasi-coherent sheaves on X. For instance, Mur-fet in [7,Remark B.7] raises the question of whether Flat(X) is closed under products,for X a noetherian scheme. This property is crucial to showing that in Ch(A), the cat-egory of unbounded chain complexes of A-modules (A commutative noetherian ring),the complex HomA(I, I ′) is a complex of flat modules, for injectives I, I ′ ∈ Ch(A). Wepoint out that the usual notion of flatness in (X) is not categorical, as it is shown in[5]. Recently Saorın and Stovıcek [8,4.2] have given a positive answer to this questionfor Dedekind schemes.

If X is affine, there is a canonical equivalence between Flat(X) and the class Flat(R)of flat R-modules, where X = Spec(R). Now if X is also Dedekind it is well knownthat Flat(R) coincides with the class of torsion-free R-modules. So, for an arbitraryscheme, it makes sense to define the class F of locally torsion-free quasi–coherentsheaves as the class of F ∈ (X) such that F(U) is a torsion-free OX(U)-module, foreach affine open set U . This class contains Flat(X) in general, and indeed it coincideswith it for Dedekind schemes. Thus this work is devoted to study the class F . Moreprecisely, we characterize the product object of a family of quasi–coherent sheaves inF obtaining, as a consequence, the forementioned result of Saorın and Stovıcek forDedekind schemes. One of the consequences of this characterization is that for anintegral scheme the class F induces a hereditary torsion theory in (X) (see also [1]for an extensive study about torsion theories in Sh(X) (the category of sheaves ofOX -modules) and (X)).



Flat covers are shown that exist in [3,Theorem 4.1]. We show the existence of coverswith respect to the class F , as well.

Keywords: torsion-free quasi–coherent sheaf, integral scheme, torsion theory, cover

2010 Mathematics Subject Classification: 13D30; 18E40; 18F20; 14F05; 18A30

Reference:

[1] Bueso, J.L., Torrecillas, B., Verschoren, A. Generalized Local Cohomology andQuasicoherent Sheaves. J. Algebra 138 (1991), 298–312.

[2] Enochs, E., Shortening filtrations. Science China Math. 55 (2012), 687–693.

[3] Enochs, E., Estrada, S. Relative homological algebra in the category of quasi-coherent sheaves. Advances in Mathematics 194 (2005), 284–295.

[4] Estrada, S., Guil Asensio, P.A., Prest, M., Trlifaj, J. Model category structuresarising from Drinfeld vector bundles. Advances Math. 231 (2012), 1417–1438.

[5] Estrada, S., Saorın, M. Locally finitely presented categories with no flat objects.Forum. Math. to appear. Available at arXiv:1204.5681.

[6] Grothendieck, A. Sur quelques points dalgebre homologique. Tohoku Math. J. 9(1957), 119–221.

[7] Murfet, D. The Mock homotopy category of projectives and Grothendieck duality.PhD thesis, Australian National University, (2007), available athttp://www.therisingsea.org/thesis.pdf.

[8] Saorın, M. Stovıcek, J. On exact categories and applications to triangulated cate-gories. Advances Math. 228 (2011), 968–1007.



2.59 Seda OGUZ

Some Finiteness Conditions of Semigroup and MonoidConstructions

Seda OGUZ1, Eylem GUZEL KARPUZ2

1 Department of Secondary School Science and Mathematics Education, CumhuriyetUniversity, Sivas, Turkey

[email protected]

2 Department of Mathematics, Karamanoglu Mehmetbey University, Karaman,Turkey

[email protected]

Abstract: A property P of semigroups (monoids) is a finiteness condition if all finitesemigroups (monoids) satisfy P. Periodicity, residually finiteness, finite generation, fi-nite presentability, locally finiteness and solvable word problem are some examplesof finiteness conditions. In this talk, firstly, I will present some important resultson finite presentability preserved under some semigroup and monoid constructions,namely Bruck-Reilly extension of a monoid, HNN extension, direct product of semi-groups. Then I will give necessary and sufficent conditions for generalized Bruck-Reilly*-extension of a group to be finitely generated and finitely presented. Finally I willpresent some related results on generalized Bruck-Reilly *-extension.

Keywords: Finiteness condition, Bruck-Reilly extension, generalized Bruck-Reilly*-extension

2010 Mathematics Subject Classification: 16S15;20F05

Reference:

[1] Howie, J. M. Fundamentals of Semigroup Theory. Clarendon Press-Oxford,(1995).

[2] Kocapinar, C., Karpuz, E. G., Ates, F., Cevik, A. S. Grobner-Shirshov bases of thegeneralized Bruck-Reilly ∗-extension. Algebra Colloquium 19 (2012), 813-820.

[3] Araujo, I.M., Ruskuc, N. Finite Presentability of Bruck-Reilly Extensions ofGroups. Journal of Algebra 242 (2001), 20-30.

[4] Araujo, I.M., Finite Presentability of Semigroup Constructions. InternationalJournal of Algebra and Computation 12 (2002), 19-31.



2.60 Salahattin OZDEMIR

Rad-supplementing Modules

Salahattin OZDEMIR

Department of Mathematics, Dokuz Eylul University, Buca, Izmir, [email protected]

Abstract: Let R be a ring and M be a left R-module. We call M (ample) Rad-supplementing if it has a (an ample) Rad-supplement in each module in which it iscontained as a submodule. If M is Rad-supplementing, then every direct summand ofM is Rad-supplementing, but not each factor module of M . Any finite direct sum ofRad-supplementing modules is Rad-supplementing. Every module with compositionseries is (Rad-)supplementing. R is left perfect if and only if R is semilocal, reducedand the free left R-module (RR)(N) is Rad-supplementing if and only if R is reducedand the free left R-module (RR)(N) is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every leftR-module is (ample) Rad-supplementing if and only if R/P (R) is left perfect, whereP (R) is the sum of all left ideals I of R such that Rad I = I.

Keywords: supplement, Rad-supplement, Rad-supplementing module, perfect ring.

2010 Mathematics Subject Classification: 16D10;16L30

Reference:

[1] Anderson, F. W. and Fuller, K. R. Rings and Categories of Modules, New-York:Springer, 1992.

[2] Buyukasik, E., Mermut, E. and Ozdemir, S. Rad-supplemented modules. Rend.Semin. Mat. Univ. Padova 124 (2010), 157–177.

[3] Clark, J., Lomp, C., Vanaja, N. and Wisbauer, R. Lifting modules, Frontiers inMathematics, Basel: Birkhauser Verlag, Supplements and projectivity in moduletheory, 2006.

[4] Lam, T. Y. Lectures on modules and rings, vol. 189 of Graduate Texts in Mathe-matics, New York: Springer-Verlag, 1999.

[5] Zoschinger, H. Moduln, die in jeder Erweiterung ein Komplement haben, Math.Scand. 35 (1974c), 267–287.



2.61 Gulcan OZKUM

Some Generalizations on Prime Gamma Rings withDerivation and Gamma Rings with Module Valued

(σ, τ)-Derivation

Gulcan OZKUM1, Muharrem SOYTURK2

1 Department of Mathematics, Kocaeli University, Umuttepe, Kocaeli, [email protected]

2 Department of Mathematics, Kocaeli University, Umuttepe, Kocaeli, [email protected]

Abstract: In this study some known results on prime rings with derivation are gen-eralized to prime Gamma rings and some known results on rings with module valued(σ, τ)-derivation are generalized to Gamma rings.

Keywords: Gamma ring, Prime ring, Derivation, Module valued (σ, τ)-derivation,One-sided ideal.

2010 Mathematics Subject Classification: 16N60;16Y99;16S99

Reference:

[1] Barnes, W. E. On the Γ-rings of Nobusawa. Pasific J. of Math. 18 (1966), no. 3,411–422.

[2] Jing, F. J. On the derivations of Γ-rings. Qufu Shifan Daxue Xuebao Ziran KexueBan. 13 (1987), no. 4, 159–161.

[3] Posner, E. Derivations in Prime Rings. Proc. Amer. Math. Soc. 8 (1957),1093–1100.

[4] Soyturk, M. The commutativity in prime gamma rings with derivation. Tr. J. ofMath. (4) 18 (1994), 149–155.

[5] Kyuno, S. On prime gamma rings. Pasific J. of Math. 75 (1978), no. 1, 185–190.

[6] Soyturk, M. On (σ, τ)-derivations with module values, Tr. J. of Math. 20 (1996),563–569.

[7] Ozkum, G. Gamma halkaları uzerinde turevler, MsC Thesis, Kocaeli University,Institute of Science, (2000)



2.62 Talat PARVEEN

Lie Ideals and Centralizing Derivations in SemiprimeRings

Talat PARVEEN

Department of Mathematics College of Science Al Jouf University, Al Jouf, Kingdomof Saudi Arabia

[email protected]

Abstract:



2.63 Manoj Kumar PATEL

FI-Semi Projective Modules and their EndomorphismRings

Manoj Kumar PATEL

[email protected]

Abstract:



2.64 Hourong QIN

K2 and L-values of elliptic curves

Hourong QIN

Department of Mathematics Nanjing University Nanjing 210093, [email protected]

Abstract: For some elliptic curves defined over the field of rational number fields, thegenerators of the torsion free part of the second K-group can be given explicitly. Forelliptic curves defined by the tempered polynomial, we establish the relation betweenMahler measures and the values of L-functions via Beilinsons regulator map.



2.65 Udhayakumar RAMALINGAM

Derived Functors of Hom Relative to n−flat Covers

Udhayakumar RAMALINGAM

Department of Mathematics, Periyar University, Salem, Tamil Nadu, INDIAudhayaram [email protected]

Abstract: n this paper, we study the derived functors of Hom that are computed byusing n-flat resolutions of Hom. These are denoted by Fnext

i and we compare thesewith the usual Exti’s.

Reference:

[1] Aldrich, S. T., Enochs, E., and Lopez Ramos, J. A. Derived functors of Homrelative to flat covers. Math Nachr. 242 (2002), 17-26.

[2] Xu, J. flat covers of modules, Lecture Notes in Mathematics, Springer, 1996.



2.66 Kulumani M. RANGASWAMY

Some Recent Advances in Leavitt Path Algebras overArbitrary Graphs

Kulumani M. RANGASWAMY

University of Colorado, Colorado Springsi, [email protected]

Abstract: In this talk, we give a complete characterization of Leavitt path algebrasover arbitrary graphs having at most finitely many non-isomorphic simple modules



2.67 Nadeem ur REHMAN

Identities with Additive Mappings in Rings

Nadeem ur REHMAN

Department of Mathematics Aligarh Muslim University Aligarh 202002, [email protected]

Abstract: Let R be an associative ring with center Z(R). Let n > 1 be an integer.A ring R is n-torsion free if nx = 0, x ∈ R, implies x = 0. The Lie product (or acommutator) of elements x, y ∈ R will be denoted by [x, y] (i.e., [x, y] = xy− yx), andthe Jordan product (or an anti-commutator) of elements x, y ∈ R will be denoted byxy (i.e., xy = xy+yx). Recall that a ring R is prime if aRb = 0, a, b ∈ R, impliesthat either a = 0 or b = 0. Furthermore, a ring R is called semiprime if aRa = 0,a ∈ R, implies a = 0. An additive mapping D : R → R is called a derivation ifD(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R. A mapping f of Rinto itself iscalled centralizing on R if [f(x), x] ∈ Z(R) holds for all x ∈ R, in the special casewhen [f(x), x] = 0 holds for all x ∈ R, the mapping f is said to be commuting onR. A result of Bresar [2], which states that every additive commuting mapping of aprime ring R is of the form x 7→ λ(x) + η(x) where λ is an element of C, the extendedcentroid of R, and η : R → C is an additive mapping, should be mentioned. Recently,A. Fosner and Vukman [5] have proved the following result: let R be a 2torsion freesemiprime ring. Suppose that an additive mapping f : R → R satisfies the relation[f(x), x2] = 0 for all x ∈ R. In this case f is commuting on R. The result above hasbeen proved by Bresar and Hvala in case R is a prime ring [4]. In the present talk ourobjective to obtaine the following result, which fairly generalizes the above mentionresult. Let n > 1 be an integer and let R be a n!-torsion free semiprime ring. Supposethat f : R → R is an additive mapping satisfying the relation [f(x), xn] = 0 for allx ∈ R. Then f is commuting on R. Moreover, we will also write some interestingconsequences of this result.

Reference:

[1] K.I. Beidar, W.S. Martindale III, A.V. Mikhalev: Rings with generalized identities,Marcel Dekker, Inc. New York 1996.

[2] M. Bresar: Commuting traces of biadditive mappings, commutativity- preservingmappings and Lie mappings, Trans. Amer. Math. Soc., 335 (1993), 525-394.

[3] M. Bresar: On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47(1993), 291-296.

[4] M. Bresar, B. Hvala: On additive maps of prime rings, Bull. Austral. Math. Soc.51 (1995), 377-381.

[5] A. Fosner, J. Vukman: Some results concerning additive mappings and derivationson semiprime rings, Publ. Math. Debrecen, 78 (2011), 575-581.



2.68 P. G. ROMEO

Von Neumann Algebras and Semigroups

P. G. ROMEO

Department of Mathematics, CUSAT, Indiaromeo [email protected]

Abstract:



2.69 Mohammad SADEK

Formal Groups, Elliptic Curves, and Modular Forms

Mohammad SADEK

Department of Mathematics and Actuarial Science, American University in Cairo,Cairo, Egypt

[email protected]

Abstract: To any elliptic curve E defined over the rational field, one can associatea formal group. We show how one can use the latter algebraic structure to obtainanalytic and combinatorial results. For example, we evaluate some of the coefficientsof the L-series attached to E, and illustrate many factorization results for binomialcoefficients over prime fields.

Keywords: Formal Groups, Elliptic Curves.

2010 Mathematics Subject Classification: 14L05;14H52



2.70 Ali M. SAGER

On Prime Near-rings with Generalized (σ, τ)- derivations

Ali M. SAGER

Department of Mathematical Sciences University of Tripoli [email protected]

Abstract: The main object of this paper is result due to Golbasi reads as follows:Let N be a prime near-ring with nonzero generalized (σ, τ)-derivation f associatedwith (σ, τ)-derivation such that τf = fτ , σf = fσ and dσ = σd, τd = dτ . If[f(N), f(N)] = 0, then(N,+) is abelian. Moreover, if N is 2- torsion free, thenN is acommutative ring.

Here we should mention that the proof given in [5] was not correct. (At one pointboth left and right distributivity were assumed). Our present paper may be regardedas a correction to that proof.



2.71 Liang SHEN

On Small Dual Rings

Liang SHEN

Department of Mathematics, Southeast University, Nanjing 210096, [email protected]

Abstract: A ring R is called right (small) dual if every (small) right ideal of R is aright annihilator. Left (small) dual rings can be defined similarly. And a ring R iscalled (small) dual if R is left and right (small) dual. It is proved that R is a dualring if and only if R is a semilocal and small dual ring. Several known results aregeneralized and properties of small dual rings are explored. As applications, somecharacterizations of QF rings are obtained through small dualities of rings.

Keywords: Small dual rings, dual rings, semilocal rings, QF rings.

2010 Mathematics Subject Classification: 16D25; 16L60

Reference:

[1] Anderson, F. W., Fuller, K. R. Rings and Categories of Modules. 2nd ed, NewYork: Springer-Verlag, 1992.

[2] Baer, R. Rings with duals. Amer. J. Math. 65 (1943), 569–584.

[3] Dung, N. V., Huynh, D. V., Wisbauer, R. Quasi-injective modules with acc or dccon essential submodules. Arch. Math. 53 (1989), 252–255.

[4] Hajarnavis, C. R., Norton, N.C. On dual rings and their modules. J. Algetra 93(1985), 253–266.

[5] Hall, M. A type of algebraic closure. Ann. of Math. 40 (1939), 360–369.

[6] Kaplansky, I. Dual rings. Ann. of Math. 49 (1948), 689–701.

[7] Koike, K. Dual rings and cogenerator rings. Math. J. Okayama Univ. 37 (1995),99–103.

[8] Lam, T. Y. A First Course in Noncommutative Rings. New York: Springer-Verlag,1991.

[9] Lomp, C. On semilocal modules and rings. Comm. Algebra (4) 27 (1999), 1921–1935.

[10] Morita, K. On S-rings in the sense of F. Kasch. Nagoya Math. J. 27 (1966),687-C695.

[11] Nicholson, W. K., Yousif, M. F. On perfect simple-injective rings. Proc. Amer.Math. Soc. (4) 125 (1997), 979–985.

[12] Nicholson, W. K., Yousif, M. F. Quasi-Frobenius Rings. Cambridge Tracts inMathematics 158, Cambridge University Press, 2003.

[13] Page, S. S., Zhou, Y. Q. Quasi-dual rings. Comm. Algebra (1) 28 (2000), 489-504.



[14] Rutter, E. A. Rings with the principle extension property. Comm. Algebra (3)3 (1975), 203-212.

[15] Shen, L. A note on quasi-Johns rings. Contemporary Ring Theory 2011, Proceed-ings of the Sixth China-Japan-Korea International Conference on Ring Theory,(2012), 89–96.

[16] Shen, L. J-regular rings with injectivities. Algebta Colloq. (2) 20 (2013), 343-347.

[17] Shen, L., Chen, J.L. New characterizations of quasi-Frobenius rings. Comm.Algebra 34 (2006), 2157–2165.

[18] Yousif, M. F. On continuous rings. J. Algebra 191 (1997), 495–509.

[19] Zhou, D., Li, D., Guo, L. Annihilator conditions relative to a class of modules.Thai J. Math. (3) 8 (2010), 419-428.



2.72 Faiza SHUJAT

Characterizations of Generalized Derivation in Rings

Faiza SHUJAT

Department of Mathematics Aligarh Muslim University, [email protected]

Abstract: Let R be a ring. An additive mapping F : R → R is said to be a generalizedderivation if there exists a derivation d : R → R such that F (xy) = F (x)y + xd(y),for all x, y ∈ R. In [Canad Math Bull. 22(4) (1979), 509-511] Herstein determined thestructure of a prime ring R admitting a nonzero derivation d such that the values ofd commute, that is d(x)d(y) = d(y)d(x) for all x, y ∈ R. Perhaps even more naturalmight be the question of what can be said of a derivaion when elements in a primering commute with all the values of a nonzero derivation. Herstein addressed thisquestion by proving the following result: If d is a nonzero derivation of a prime ringR and a /∈ Z(R) is such that [a, d(x)] = 0 for all x ∈ R, then R has characteristic 2, a2 ∈ Z(R) and d(x) = [λa, x], for all x ∈ R and λ ∈ C, the extended centroid ofR. Recently Albas and Nurcan [Algebra Colloq, 11 (3) (2004), 399-410] proved that ifR is a non-commutative prime ring and d is a generalized derivation with associatedderivation α of R and for all x ∈ R, [a, d(x)] = 0, then either a ∈ C or there existλ, η ∈ C such that d(x) = ηx + λ(ax + xa) for all x ∈ R. In the present note wecharacterize the generalized derivation F of a prime (semiprime) ring satisfying theabove mentioned condition. Our theorems extend results of Bresar [J. Algebra 156(1993), 385-394, Lemma 2.2], Aydin [Intern J. Algebra 5 (1) (2011), 17-23, Theorem2.3], Nurcan [Algebra Colloq, 13 (3) (2006), 371-380, Lemma 4.2] and a result ofHerstein [Canad Math Bull. 22(4) (1979), 509-511, Theorem].

Keywords: Prime and semiprime rings, generalized derivation

2010 Mathematics Subject Classification: 16W25, 16N60



2.73 Ebru SOLAK

(1, 2)-groups for a Regulator Quotient of Exponent p4

Ebru SOLAK1, Otto MUTZBAUER2

1Department of Mathematics, Middle East Technical University, Ankara, [email protected]

2 Universitat Wurzburg, Mathematisches Institut, Am Hubland, 97074 Wurzburg,Germany

[email protected]

Abstract: A torsion-free abelian group G is almost completely decomposable if Gcontains a completely decomposable subgroup A with G/A a finite group. The regula-tor R(G) is the intersection of all regulating subgroups of G and is again a completelydecomposable subgroup of finite index in G.Let p be a prime, (1, 2) = (τ1, τ2 < τ3) a set of types, partially ordered as indicatedwith τi(p) 6= ∞. An almost completely decomposable group G is called a (1, 2)-groupif G/R(G) is p-primary and R(G) = R1 ⊕ R2 ⊕ R3 where Ri is completely decom-posable and τi-homogeneous. Such a group has a unique regulating subgroup thatcoincides with its regulator R, and, up to near-isomorphism, unique indecomposabledecompositions by the theorem of Arnold. In a joint work with Otto Mutzbauer,cf. [1] we determined the near-isomorphism classes of indecomposable (1, 2)-groups for

a regulator quotient of exponent p4 and we showed that

• there are four near-isomorphism classes of indecomposable (1, 2)-groups G withexp

(G/R(G)

)≤ p3. The regulator quotients are isomorphic to Z /p2 Z, Z /p3 Z,(

Z /p3 Z)⊕ (Z /pZ),

• there are eight near-isomorphism classes of indecomposable (1, 2)-groups G withexp

(G/R(G)

)= p4. The regulator quotients are isomorphic to Z /p4 Z,(

Z /p4 Z)⊕ (Z /pZ) and

(Z /p4 Z

)⊕

(Z /p2 Z

).

Reference:

[1] Mutzbauer, Otto and Solak, Ebru. (1,2)-Groups for a regulator quotient of expo-nent p4, in Groups and Model Theory, Contemporary Mathematics, vol. 576,Amer. Math. Soc., Providence, RI, 2012, pp 269-285.

[2] O. Mutzbauer, E. Solak, (1, 2)-groups with p3-regulator quotient, J. Alg. 320(2008), 3821-3831.

[3] D. M. Arnold, Abelian Groups and Representations of Partially Ordered Sets,CMS Advanced Books in Mathematics, Springer-Verlag, New York, 2000.

[4] A. Mader, Almost Completely Decomposable Groups, Gordon Breach, 2000.

[5] E. Solak, Almost completely decomposable groups of type (1, 2), Dissertation(Wurzburg 2007).



2.74 Carlos SONEIRA CALVO

Yetter-Drinfeld Categories in a Weak Context

Carlos SONEIRA CALVO

Department of Pedagoxıa e Didactica, University of A Coruna, [email protected]

Abstract: In this work we introduce the categories of left-left, right-right, left-rightand right-left Yetter Drinfeld modules over a weak braided Hopf algebra in a strictmonoidal category and prove that there exists a categorical equivalence between allof them. We also establish categorical equivalences by changing the weak braidedHopf algebra by its (co)opposite. In this sense, the new notions allow to recover asa particular instance the theory developed in [6] for the case of bialgebras. Finallythe general results are illustrated with an example coming from projections of weakbraided Hopf algebras.

Keywords: Weak Yang-Baxter operator, Yetter-Drinfeld module, weak (braided)Hopf algebra.

2010 Mathematics Subject Classification: 81R50;18D10;18D35;16T99.

Reference:

[1] Alonso Alvarez, J.N., Fernndez Vilaboa, J.M.; Gonzlez Rodrguez, R., Weak Hopfalgebras and weak Yang-Baxter operators. J. of Algebra 320 (2008), no. 6,2101–2143.

[2] Alonso Alvarez, J.N., Fernndez Vilaboa, J.M., Gonzlez Rodrguez, R., SoneiraCalvo, C., Projections of Weak Braided Hopf Algebras, Science China Mathe-matics 54 (2011), no. 5, 877–906.

[3] Alonso Alvarez, J.N., Fernandez Vilaboa, J.M., Gonzalez Rodrıguez, R., SoneiraCalvo, C. The monoidal category of Yetter-Drinfeld modules over a weak braidedHopf algebra, http://arxiv.org/abs/1203.2474.

[4] Alonso Alvarez, J.N., Fernandez Vilaboa, J.M., Gonzalez Rodrıguez, R., SoneiraCalvo, C. Yetter-Drinfeld categories associated to a weak braided Hopf algebra.Arabian J. of Mathematics 2 (2013) no.1, 1–18.

[5] Joyal, A., Street, R., Braided tensor categories, Adv. in Math. 102 (1993), 20–78.

[6] Radford, D.E., Towber, J. Yetter-Drinfeld categories associated to an arbitrarybialgebra. J. of Pure and Applied Algebra, 87 (1993), 259–279.



2.75 David SSEVVIIRI

2-Primal Modules

Nico J. GROENEWALD , David SSEVVIIRI

Department of Mathematics and Applied Mathematics, Nelson MandelaMetropolitan University, South Africa


Abstract: A notion of 2-primal rings is generalized to modules by defining 2-primalmodules. We show that the implications between rings which are reduced, IFP, re-versible, semi-symmetric and 2-primal are preserved when the notions are extendedto modules. Like for rings, 2-primal modules bridge the gap between modules overcommutative rings and modules over non-commutative rings; for instance, for 2-primalmodules, prime submodules coincide with completely prime submodules. Completelyprime submodules and reduced modules are both characterized. A generalization of2-primal modules is done where 2-primal and NI modules are a special case.

Keywords: 2-primal modules; prime submodules; completely prime submodules.

2010 Mathematics Subject Classification: 16D10; 16N60; 16N80; 16S90;

Reference:

[1] V. A. Andrunakievich and Ju. M. Rjabuhin, Special modules and special radicals,Soviet Math. Dokl, 3 (1962), 1790-1793. Russian original: Dokl. Akad. NaukSSSR, 147 (1962), 1274-1277.

[2] N. Argac, N. J. Groenewald, A generalization of 2-primal near-rings, QuaestionesMathematicae, 27 (2004), 397-413.

[3] J. A. Beachy, Some aspects of Noncommutative localization, NoncommutativeRing Theory, Kent State, 1975, Lecture Notes in Math. Springer-Verlag545,(1976), 231.

[4] M. Behboodi, A generalization of Baer’s lower Nilradical for modules, Journal ofAlgebra and Its Applications,World Scientific, 6 (2007), 337 - 353.

[5] M. Behboodi, On the prime radical and Baer’s lower nilradical of modules, ActaMath. Hungar., 122 (2008), 293-306.

[6] M. Behboodi, H. Koohy, Weakly prime modules, Vietnam J. Math, 32 (2004),303-317.

[7] H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral.Math. Soc., 2 (1970), 363-368.

[8] L. Bican, T. Kepka and P. Nemec, Rings, Modules and Preradicals, Lecture notesin pure and applied mathematics no.75, Marcel Dekker Inc., New York, 1982.

[9] G. F. Birkenmeier, H. E. Heatherly, and E. K. Lee, Completely prime ideals andassociated radicals, in: S.K. Jain, S.T. Rizvi (Eds), Proceedings of the BiennialOhio State-Denison Conference, 1992, World Scientific, Singapore, 1993, 102-129.



[10] W. D. Blair, H. Tsutsui, Fully prime rings, Comm. Algebra, 22 (1994), 5389-5400.

[11] P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31 (1999) 641-648.

[12] R. C. Courter, Fully idempotent rings have regular centroids, Proc. Amer. Math.Soc., 43 (2), (1974), 293-296.

[13] R. C. Courter, Rings all of whose factor rings are semiprime, Canad. Math.Bull., 12, (1969), 417-426.

[14] J. Dauns, Prime modules, Reine Angew. Math. 298 (1978), 156-181.

[15] B. de la Rosa, S. Veldsman, A relationship between ring radicals and moduleradicals, Quaestiones Mathematicae, 17 (1994), 453-467.

[16] M. Hongan, On strongly prime modules and related topics, Math. J. OkayamaUniv., 24 (1982), 117- 132.

[17] U. Hwang, Y. C. Jeon, Y. Lee, Structure and topological conditions of NI rings,J. Algebra. 302 (2006), 186-199.

[18] N. K. Kim, Y. Lee, Extensions of reversible rings, J. Pure and Applied Algebra,185 (2003), 207-223.

[19] J. Lambek, On the presentation of modules by sheaves of factor modules, Canad,Math. Bull. 14 (1971), 359-368.

[20] T. K. Lee, Y. Zhou, Reduced modules, Rings, Modules, Algebra and Abelian group.Lectures in Pure and Appl Math 236, 365-377, Marcel Decker, New York, 2004.

[21] G. Marks, On 2-primal ore extensions, Comm. Algebra, 29 (2001), 2113-2123.

[22] G. Marks, Reversible and symmetric rings, J. Pure and Applied Algebra, 174(2002) 311-318.

[23] G. Marks, A taxonomy of 2-primal rings, J. Algebra, 266 (2003), 494-520.

[24] Y. V. Reddy, C. V. L. N. Murty, Semi-symmetric ideals in near-rings, Indian J.Pure Appl. Math. 16 (1985), 17-21.

[25] M. B. Rege, A. M. Buhphang, On reduced modules and rings, Int Electronic J.Algebra. 3 (2008), 58-74.

[26] G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans.Amer. Math. Soc., 184 (1973), 43-60.

[27] D. Ssevviiri, N. J. Groenewald, Classical completely prime submodules, submittedfor publication.

[28] D. Ssevviiri, N. J. Groenewald, Completely prime submodules, submitted forpublication.



2.76 Serap SAHINKAYA

δ-semiregular Endomorphism Rings and the Dual ofYamagata’s Theorem

Serap SAHINKAYA

Department of Mathematics, Gebze Institute of Technology Cayirova Campus,41400, Gebze- Kocaeli, Turkey

[email protected] , [email protected]

Abstract: Recently, Lee and Zhou [1] characterized a module for which S/∇ is reg-ular, where ∇ = f ∈ End(M) | f(M) ≪ M and idempotents lift modulo ∇, whichis called the dual Yamagat’s theorem.

In this paper, we obtained dual Yamagata’s theorem on∇δ = f ∈ End(M) | f(M) ≪δ MWe also characterize a module whose endomorphism ring is δ-semiregular.

Keywords: Regular ring, δ-semiregular module, δ-semipotent module.

2010 Mathematics Subject Classification: 16D40; 16E50; 16N20

Reference:

[1] Lee, T. K., Zhou, Y. (2011). On (strong) lifting of idempotents and semiregularendomorphism rings. Colloq. Math. 125: 99-113.



2.77 Yahya TALEBI

CP -Modules

Yahya TALEBI, Ali Reza Moniri HAMZEKOLAEE and Mehrab HOSSEINPOUR

Department of Mathematics, University of Mazandaran, Babolsar, [email protected]

[email protected]@umz.ac.ir

Abstract: Let R be a ring and M be an R-module. We say that M is CP (SCP )provided every (simple) cosingular R-module is M -projective. We investigate somegeneral properties of these new classes of modules. We also give some equivalentconditions for a (SCP ) CP -module. It is showon that every simple cosingular R-module is projective if and only if R is a GV -ring (GCO-ring).

Keywords: CP -module, SCP -module, cosingular-module, projective module.

2010 Mathematics Subject Classification: 16D90; 16D99



2.78 Bertha TOME-ARREOLA

Decomposing Modules into Direct Sums of CotypeSubmodules

Alejandro Alvarado-Garcıa1, Hugo Alberto Rincon-Mejıa1, Jose Rıos-Montes2 andBertha Tome-Arreola1

1 Facultad de Ciencias, Universidad Nacional Autonoma de Mexico, Ciudad deMexico, Mexico

[email protected]

2 Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Ciudad deMexico, Mexico

[email protected]

Abstract: We study decompositions of an amply supplemented module M into adirect sum of cotype submodules. For this purpose we introduce some conaturalclasses, such as the class of all q-molecular modules, the class of all q-bottomlessmodules, the class of all q-discrete modules and the class of all q-continuous modules,which are defined for every ring. We also find conditions on the structure of the cotypesubmodules of an amply supplemented module M, in order for M to have certain directsum decompositions.

Keywords: Conatural class, cotype submodule, supplemented module

2010 Mathematics Subject Classification: 16D80

Reference:

[1] Alvarado-Garcıa, A., Rincon-Mejıa, H., Rıos-Montes, J. On the lattices of naturaland conatural classes in R-mod, Comm. Algebra, 29(2) (2001), 541-556.

[2] Alvarado-Garcıa, A., Rincon-Mejıa, H., Rıos-Montes, J. On some lattices of mod-ule classes, J. Algebra Appl., 5(1) (2006), 105-117.

[3] Alvarado-Garcıa, A., Rincon-Mejıa, H., Rıos-Montes, J., Tome-Arreola, B. Onconatural classes and cotype submodules, IEJA, Volume 11 (2012) 64-81.

[4] Clark, J., Lomp, C., Vanaja, N. and Wisbauer, R. Lifting Modules. Supplementsand Projectivity in Module Theory, Frontiers in Mathematics, Birkhauser Ver-lag, Bassel, 2006.

[5] Dauns, J., Zhou, Y. Classes of modules, Pure and Applied Mathematics (BocaRaton), 281. Chapman & Hall/CRC, Boca Raton, FL, 2006.

[6] Kasch, F. Modules and Rings, Academic Press Inc. (London) Ltd, 1982.

[7] Mohamed, S.H., Muller, B. Continuous and Discrete Modules, London Math. Soc.Lectures Notes 147, Cambridge Univ. Press, 1990.

[8] Wisbauer, R. Foundations of Module and Ring Theory: A handbook for studyand research, Gordon and Breach Science Publishers, 1991.



2.79 Hamid USEFI

Isomorphism Problem for Enveloping Algebras

Hamid USEFI

Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s, NL, Canada, A1C 5S7

[email protected]

Abstract: Let L be a Lie algebra with universal enveloping algebra U(L). A particularinvariant of L is said to be determined by U(L) if every Lie algebra H also possessesthis invariant whenever U(L) and U(H) are isomorphic as associative algebras. Thus,roughly speaking, an invariant of L is determined by U(L) whenever it can be deducedfrom the algebraic structure of U(L) without any direct knowledge of the underlyingLie algebra L itself. For example, it is well-known that the dimension of a finite-dimensional Lie algebra L is determined by U(L) since it coincides with the Gelfand-Kirillov dimension of U(L). In this talk we demonstrate that certain other invariantsof L are also determined by U(L). We also discuss the most far reaching problem ofthis sort that asks whether or not (the isomorphism type of) every Lie algebra L isdetermined by U(L).



2.80 Burcu UNGOR

On the Pure-Injectivity Profile of a Ring

Abdullah HARMANCI1, Sergio R. LOPEZ-PERMOUTH2 and Burcu UNGOR3

1 Department of Mathematics, Hacettepe University, Ankara, [email protected]

2 Department of Mathematics, Ohio University, Athens, OH, [email protected]

3 Department of Mathematics, Ankara University, Ankara, [email protected]

Abstract: An analog of the injective profile of a ring introduced by Lopez-Permouthand Simental [3], with relative injectivity replaced by relative pure-injectivity, is in-vestigated. Emphasis is placed on comparing and contrasting the properties of theinjective profile and the pure-injective profile introduced here. In particular, the ana-log notion of poor modules ([1], [2]) in this context is considered and properties ofpure-injectively poor modules are determined. Also, some characterizations of rightpure-semisimple rings are obtained in terms of the notion of pure-injectively poormodules.

Keywords: Poor module, pure-injectively poor module, pure-injectivity domain,pure-semisimple ring.

2010 Mathematics Subject Classification: 13C99; 16D80; 16U80

Reference:

[1] Alahmadi, A. N., Alkan, M., Lopez-Permouth, S. R. Poor modules: The oppositeof injectivity. Glasgow Math. J. 52(A) (2010), 7-17.

[2] Er, N., Lopez-Permouth, S. R., Sokmez, N. Rings whose modules have maximaland minimal injectivity domains. J. Algebra 330 (2011), 404-417.

[3] Lopez-Permouth, S. R., Simental, J. Characterizing rings in terms of the extent ofthe injectivity and projectivity of their modules. J. Algebra 362 (2012), 56-69.



2.81 Lia VAS

A Class of Rings with Dimension

Lia VAS

University of the Sciences, 600 S. 43rd St., Philadelphia, PA 19104, [email protected]

Abstract: Inspired by the existence of a well-behaved dimension of a finite von Neu-mann algebra, we study assumptions on a ring that assure the existence of a well-behaved dimension. We demonstrate that these assumptions are satisfied for stronglysemihereditary rings with a positive definite involution. This extends the class of finiteBaer ∗-rings previously known to have a dimension since the extended class includessome rings that are not Rickart ∗-rings.



2.82 Indah Emilia WIJAYANTI

On Fully Prime Radicals

Indah Emilia Wijayanti, Dian Ariesta Yuwaningsih

Department of Mathematics Universitas Gadjah Mada, Indonesiaind [email protected] and ind [email protected]

Abstract: In this paper we give a further study on fully prime submodules. For anyfully prime submodules we define a product called -product. The investigation furtherof fully prime submodules in this work is related to this product. Moreover we alsointroduce the fully prime radicals of submodules. We show that the fully prime radicalof any submodule can be characterize by the m-system. As a special case, the fullyprime radical of a module M is the intersection of all minimal fully prime submodulesof M .

Keywords: Fully invariant submodules, fully prime submodules, fully prime radicalsof submodule, fully prime radical of module



2.83 Feyza YALCIN

Complex Fibonacci p-Numbers

Dursun TASCI, Feyza YALCIN

Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, [email protected] and [email protected]

Abstract: In the present paper, the complex Fibonacci p-numbers are defined byextending the definition of complex Fibonacci numbers at Gaussian integers. Then,three-dimensional version of the Fibonacci p-numbers is introduced.

Keywords: Complex Fibonacci p-numbers, complex Fibonacci numbers, Fibonaccip-numbers

2010 Mathematics Subject Classification: 11B39

Reference:

[1] G. Berzsenyi, Gaussian Fibonacci numbers, The Fibonacci Quarterly 15 (1977),233–236.

[2] C. J. Harman, Complex Fibonacci numbers, The Fibonacci Quarterly 19 (1981),82–86.

[3] Koshy, T. Fibonacci and Lucas Numbers with Applications. A Wiley-IntersciencePublication, New York-Wiley (2001).

[4] A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer.Math. Monthly 70 (1963), 289–291.



2.84 Ece YETKIN

A Study on 2-Absorbing Elements of C-lattices

Ece YETKIN1, Chillumuntala JAYARAM2, Unsal TEKIR1

1 Department of Mathematics, Marmara University, Goztepe, Istanbul, [email protected] and [email protected]

2 Department of Mathematics, The University of the West Indies, Bridgetown,Barbados

[email protected]

Abstract: In this paper a new type of elements in multiplicative lattices is definedwhich is called a 2-absorbing element. Some properties of this type of elements areestablished and also new characterizations for some special multiplicative lattices interms of 2-absorbing elements are obtained.

Keywords: 2-absorbing element, weakly 2-absorbing element

2010 Mathematics Subject Classification: 06F10, 06F05, 13A15

Reference:

[1] Alarcon, F. and Anderson, D.D. Commutative semirings and their lattices of ideals,Houston Journal of Mathematics, 20 (1994), 571-590.

[2] Alarcon, F. , Anderson, D.D. and Jayaram C. Some results on abstract commuta-tive ideal theory, Periodica Mathematica Hungarica, 30 (1995), 1-26.

[3] Anderson, D.D. Abstract commutative ideal theory without chain condition, Al-gebra Universalis, 6 (1976), 131-145.

[4] Anderson, D.D. and Jayaram, C. Regular lattices, Studia Scientiarum Mathemati-carum Hungarica, 30 (1995), 379-388.

[5] Anderson, D.D. and Jayaram, C. Principal element lattices, Czechoslovak Mathe-matical Journal, 46 (1996), 99-109.

[6] Anderson, D.D. and Johnson, E.W. Dilworth’s principal elements, Algebra Uni-versalis, 36 (1996), 392-404.

[7] Anderson, D.D. and Smith, E. Weakly prime ideals, Houston Journal of Mathe-matics, 29 (2003), 831-840.

[8] Badawi, A. On 2-absorbing ideals of commutative rings, Bull. Austral. Math.Soc., 75 (2007), 417-429.

[9] Badawi, A. and Darani, A. Y. On weakly 2-absorbing ideals of commutative rings,Houston Journal of Mathematics, (in press).

[10] Callıalp, F., Jayaram, C. and Tekir, U. Weakly prime elements in multiplicativelattices, Communications in Algebra, 40 (2012), 2825-2840.



[11] Dilworth, R.P. Abstract commutative ideal theory, Pacific Journal of Mathemat-ics, 12 (1962), 481-498.

[12] Jayaram, C. and Johnson, E.W. s-prime elements in multiplicative lattices, Pe-riodica Mathematica Hungarica, 31 (1995), 201-208.

[13] Jayaram, C. and Johnson, E.W. Strong compact elements in multiplicative lat-tices, Czechoslovak Mathematical Journal, 47 (122) (1997), 105-112.

[14] Jayaram, C. Primary elements in Pr··ufer lattices, Czechoslovak Mathematical

Journal, 52 (127) (2002), 585-593.

[15] Johnson J.A. and Sherette, G.R. Structural properties of a new class of CM-lattices, Canadian Journal of Mathematics, 38 (1986), 552-562.

[16] Thakare, N.K., Manjarekar C.S. and Maeda, S. Abstract spectral theory II, Min-imal characters and minimal spectrums of multiplicative lattices, Acta. Sci.Math. (Szeged)., 52 (1988), 53-67.



2.85 Erol YILMAZ

Envelopes and Weakly Radicals of Submodules

Erol YILMAZ1, Sibel CANSU2

1 Department of Mathematics, Abant Izzet Baysal University, Bolu, Turkeyyilmaz [email protected]

2 Department of Mathematics, Istanbul Technical University, Istanbul, [email protected]

Abstract: hroughout this paper all rings are commutative with identity and all mod-ules are unitary.

The radical√I of an ideal I ⊂ R is characterized as the the set of elements a ∈ R

such that an ∈ I for some positive integer n. The concept of envelope of a submoduleis the generalization of this characterization to the modules. If N is a submodule ofan R-module M , then the envelope of N in M is defined to be the set

EM (N) = rm : r ∈ R,m ∈ M and rkm ∈ N for some k ∈ Z+.

The submodule generated by the envelope is called (Baers) lower nilradical and denotedby 〈EM (N)〉. Although some methods for computing of radical of a submodule, whichdefined to be intersection of prime submodules containing N , are given in [4] and[5], it seems there is no description for the computation of the lower nilradical of asubmodule in the literature. We give a formula for the computation of 〈EM (N)〉 if aminimal primary decomposition of N is known.

A proper submodule N of an R-module M is called a weakly prime submodule if foreach m ∈ M and a, b ∈ R; abm ∈ N implies that am ∈ N or bm ∈ N . A propersubmodule N of an R-module M is called a weakly primary submodule if abm ∈ Nwhere a, b ∈ R and m ∈ M , then either bm ∈ N or akm ∈ N for some k ≥ 1. Theconcepts of weakly prime and weakly primary submodules are introduced a few yearsago and they have been studied by some authors (for example see [1], [2] and [3] ).We investigate relations between weakly prime submodules and their envelopes. Wealso give an example to show a conjecture given in [3] is false.

The weakly radical of a submodule N of M , denoted by wradM (N), is defined to bethe intersection of all weakly prime submodules containing N . Using relation betweenenvelopes and weakly prime submodules, we try to compute radical of a submoduleat least in some cases.

Keywords: Envelopes, weakly prime submodules and weakly radicals

2010 Mathematics Subject Classification: 13E05; 13E15; 13C99; 13P99



Reference:

[1] A. Azizi Weakly Prime Submodules and Prime Submodules, Glaskow Math. J. 48,343–346 (2006).

[2] A. Azizi Radical Formula and Weakly Prime Submodules, Glaskow Math. J. 51,405–412 (2009).

[3] M. Baziar, M. Behboodi, Classical Primary Submodules and Decomposition Theoryof Modules, J. of Algebra and It’s App. 8(3), 351-362 (2009).

[4] A. Marcelo, C. Rodriguez,Radical of Submodules and Symmetric Algebra, Comm.Algebra, 28(10), 4611–4617 (2000).

[5] R. L. McCasland, P. F. Smith, Generalised Associated Primes and Radicals ofSubmodules, Int. Elec. Jour. of Algebra 4 159–176 (2008).

[6] R. Y. Sharp, Steps in Commutative Algebra, Cambridge Univ. Press (1990).



2.86 Utku YILMAZTURK

Character Degrees and Derived Length

Utku YILMAZTURK

Department of Mathematics, University of Istanbul, Istanbul, [email protected]

Abstract: In this study we have obtained some sufficient conditions for the TaketaInequality namely dl(G) ≤| cd(G) | for finite solvable groups G where dl(G) denotesthe derived length of G and cd(G) denotes the set of irreducible character degree ofG.

Keywords: Character degrees, solvable groups, derived length


Reference:

[1] Gluck, D.,: Bounding the number of character degrees of a solvable group, J. Lon-don Math. Soc. 31 (2) (1985) 457–462.

[2] Isaacs, I.M.,: Groups having at most three irreducible character degrees , Proc.Amer. Math. Soc. 21, 185-188 (1969)

[3] Isaacs, I.M., Knutson, G.: Irreducible character degrees and normal subgroups, J.Algebra, 199 302-326, (1998)

[4] Isaacs, I.M.,: Character Theory of Finite Groups,AMS Chelsea Publishing, Prov-idence, RI, 2006.

[5] Lewis, M., : Derived lengths of solvable groups having five irreducible characterdegrees I, Algebras Represent. Theory 4 (2001) 469-489



2.87 Figen YUZBASI ERYILMAZ

Generalization of ⊕− δ− Supplemented Modules

Figen YUZBASI ERYILMAZ, Senol EREN

Department of Mathematics, Ondokuz Mayıs University, Atakum, Samsun,[email protected] , [email protected]

Abstract: Let R be a ring and M be a left R−module. We say that an R−module Mis a generalized ⊕−δ−supplemented module if every submodule of M has a generalizedδ−supplement which is a direct summand of M . In this paper, several properties ofthese modules are given. We showed that any finite direct sum of generalized ⊕ −δ−supplemented modules is a generalized ⊕−δ−supplemented module and every directsummand of a UC-extending generalized ⊕− δ−supplemented module is a generalized⊕− δ−supplemented.

Keywords: δ−small submodule, ⊕ − δ−supplemented module, generalized ⊕ −δ−supplemented module

2010 Mathematics Subject Classification: 16D10; 16D90; 16D99; 16N80

Reference:

[1] Clark, J., Lomp, C., Vajana, Wisbauer, R. Lifting Modules. Birkhauser Verlag.2006.

[2] Harmancı, A., Keskin, D., Smith, P. F. On ⊕−supplemented modules. Acta Math.Hungar. 83 (1999), no. 1-2, 161–169.

[3] Idelhadj, A.,Tribak, R. On sum properties of ⊕−supplemented modules. Int. J.Math. Sci. 69 (2003), 4373-4387.

[4] Keskin, D., Smith P.F., Xue. W. Rings whose modules are ⊕-supplemented. J.Algebra. 218 (1999), no.2, 470-487.

[5] Kosan, M.T. δ−lifting and δ−supplemented modules. Algebra Colloq. 14 (2007),no.1, 53-60.

[6] Mohamed, S.H., Muller B.J. Continuous and Discrete Modules. Cambridge Uni-versity Press, 1990.

[7] Ozcan, A.C., Harmancı A., Smith, P.F. Duo modules. Glasgow Math. J. 48(2006), no.3, 533-545.

[8] Smith, P.F. Modules for which every submodule has a unique closure. 1993. InRing Theory, World Sci., Singapore, 302-313.

[9] Talebi Y., Talaee, B. On generalized δ−supplemented modules. Vietnam J. Math.37 (2009), no.4, 515-525.

[10] Talebi, Y., Hamzekolaei, A.R., Tutuncu, D.K. On Rad-⊕-supplemented modules.Hadronic J. 32 (2009), no.5, 505-512.



[11] Talebi Y., Mahmoudi, A.R. On Rad-⊕-supplemented modules. Thai J. Math. 9(2011), no.2, 373–381.

[12] Ungor, B., Halıcıoglu, S., Harmancı, A. On a class of δ−supplemented modules.Bull. Malays. Math. Sci. Soc.. (accepted).

[13] Wang Y., Ding, N. Generalized supplemented modules. Taiwanese J. Math. 10(2006), no.6,1589-1601.

[14] Wang, Y. δ−small submodules and δ−supplemented modules. Int. J. Math. andMath. Sci., (2007).Article ID 58132, 8 pages, doi:10.1155/2007/58132.

[15] Wisbauer, R. Foundations of Module and Ring Theory. Gordon and Breach.Philadelphia. 1991.

[16] Xue, W. Characterizations of semiperfect and perfect rings. Publ. Mat. 40(1996), no.1, 115-125.

[17] Zhou, Y. Generalizations of perfect, semiperfect, and semiregular rings. AlgebraColloq. 7 (2000), no.3, 305-318.



2.88 Parvaneh ZOLFAGHARI

About Automorphisms Groups of Dihedral Group D9

Parvaneh ZOLFAGHARI

Department Of Mathematics Yerevan State University, Armeniaparvane [email protected]

Abstract: In this paper dihedral group D9,especially subgroups of order 6 and allautomorphisms of dihedral group D9 is being described. The main goal is to describeall hypergroups HM of order 3, which arises from dihedral group D9.

Keywords: hypergroup,dihadral group,automorphism

2010 Mathematics Subject Classification: Primary:20N20, Secondary:20D45

Reference:

[1] Dalalyan S.H., On hypergroups, prenormal subgroups and the simplest groups.Conf. dedicated to 90-anniversary of M.M. Jrbashyan, Yerevan, 2008, p. 12-14.(rus.)

[2] Dalalyan S. H., On hypergroup category, Conf, dedicated to 90-anniversary ofYSU, Yerevan, 2009, p. 8-9. (rus.)

[3] Dalalyan S. H., The equivalences between the categories of hyperobjects over groupand the corresponding categories of objects with two-side action of a group andwith an exact representation. Proc. of the fourth Int. Group Theory Conf. ofIran, Isfahan, 7-9 March 2012., p. 50-53.

[4] Dalalyan S. H., The reducibility theory of hypergroups over group, Conf. of AMUdedicated to 1400 anniversary of Anania Shirakatsi. Yerevan, 2012, p. 22-24.(rus)

[5] Zolfaghary P., The hypergroups of order 3, arising from symmetric group S3.Proc. Of the fourth Int. Group Theory Conf. of Iran, Isfahan, Isfahan, 7-9March 2012, p.198-201.

[6] Zolfaghary P., About Order 3 hypergroups Over Group Which arise From Groupsof Order 18. Math, Problems of Comp. Sci. 2012, p. 93-94.

[7] Coxeter H. S. M., Moser W. O. J., Generators and relations for discrete groups,Springer Verlag, 1972.


List of Participants


3 List of Participants

1. Jawad ABUHLAIL Saudi Arabia

2. Emine ALBAS Turkey

3. Shakir ALI India

4. Abdullah ALJOUIEE Saudi Arabia

5. Abdollah ALHEVAZ Iran

6. Mustafa ALKAN Turkey

7. Ahmad AL KHALAF Saudi Arabia

8. Dilara ALTAN Turkey

9. Meltem ALTUN Turkey

10. Abu Zaid ANSARI India

11. Nurcan ARGAC Turkey

12. Bertha TOM-ARREOLA Mxico

13. Mara Jos ARROYO Mxico

14. Orest ARTEMOVYCH Poland

15. Umamaheswaran ARUNACHALAM India

16. Mohammad ASHRAF India

17. Firat ATES Turkey

18. Serkan AYDIN Turkey

19. Yildiz AYDIN Turkey

20. Pınar AYDOGDU Turkey

21. Gonca AYIK Turkey

22. Hayrullah AYIK Turkey

23. Yasemin AYVALIK Turkey

24. Ayman BADAWI United Arab Emirates

25. Metin BAGDAT Turkey

26. Evgenii BASHKIROV Turkey

27. Samruam BAUPRADIST Thailand

28. Firdhousi BEGAM India

29. Vijay Kumar BHAT India

30. Gary F. BIRKENMEIER USA

31. Leonid BOKUT Russia

32. Kenneth A. BROWN UK

33. Engin BUYUKASIK Turkey

34. Carlos Soneira CALVO Spain

35. Ismail Naci CANGUL Turkey

36. Paula CARVALHO Portugal



37. Secil CEKEN Turkey

38. Canan CELEP YUCEL Turkey

39. Ozge CELIK Turkey

40. Ahmet Sinan CEVIK Turkey

41. Jianlong CHEN China

42. Yuqun CHEN China

43. John CLARK New Zealand

44. Merve COLPAN Turkey

45. Nadeem Ahmad DAR India

46. Kinkar Ch. DAS Republic of Korea

47. Ali Ahmed DAW Libya

48. Fatemeh DEHGHANI ZADEH Iran

49. Bilal DEMIR Turkey

50. Ismail DEMIR Turkey

51. Cagri DEMIR Turkey

52. Musa DEMIRCI Turkey

53. Nanqing DING China

54. Yilmaz DURGUN Turkey

55. Ozgur EGE Turkey

56. Ali Bulent EKIN Turkey

57. Ahmet EMIN Turkey

58. Noyan ER USA

59. Temha ERKOC Turkey

60. Pınar EROGLU Turkey

61. Nil ORHAN ERTAS Turkey

62. Alberto FACCHINI Italy

63. Sehmus FINDIK Turkey

64. Silvia Claudia GAVITO Mxico

65. Neslihan GIRGIN Turkey

66. Hanieh GOLMAKANI Iran

67. Mouloud GOUBI Algeria

68. Nico GROENEWALD South Africa

69. Serpil GUNGOR Turkey

70. Orhan GURGUN Turkey

71. Eylem GUZEL KARPUZ Turkey

72. Claus HAETINGER Brazil

73. Khalid Ali Mohammad HAMDIN India



74. Ayazul HASAN India

75. Can HATIPOGLU Portugal

76. Sebahattin IKIKARDES Turkey

77. Ilker INAM Turkey

78. Surender K. JAIN USA

79. Tekgul KALAYCI Turkey

80. Berke KALEBOGAZ Turkey

81. Yeliz KARA Turkey

82. Tolga KARAYAYLA Turkey

83. Gizem KAFKAS Turkey

84. Abidin KAYA Turkey

85. Fatma KAYNARCA Turkey

86. Derya KESKIN TUTUNCU Turkey

87. Sabri KHADIDJA Algeria

88. Shahoor KHAN India

89. Recep KILIC Turkey

90. Sibel KILICARSLAN CANSU Turkey

91. Gonca KIZILASLAN Turkey

92. Canan KOCAPINAR Turkey

93. M. Tamer KOSAN Turkey

94. Berna KOSAR Turkey

95. Irem KUCUKOGLU Turkey

96. Yeliz KURTULDU Turkey

97. Yosum KURTULMAZ Turkey

98. Semra KUS Turkey

99. Christian LOMP Portugal

100. Sergio R. LOPEZ-PERMOUTH USA

101. Adolf MADER Germany

102. Jules Clement MBA South Africa

103. Engin MERMUT Turkey

104. Pinar METE Turkey

105. Phool MIYAN India

106. Jos Ros MONTES Mxico

107. Najat Mohammed MUTHANA Saudi Arabia

108. Otto MUTZBAUER Germany

109. W. Keith NICHOLSON Canada

110. Burcu NISANCI TURKMEN Turkey



111. Sinem ODABASI Turkey

112. Seda OGUZ Turkey

113. Barbara L. OSOFSKY USA

114. Ayse Cigdem OZCAN Turkey

115. Salahattin OZDEMIR Turkey

116. Gulcan OZKUM Turkey

117. Ali PANCAR Turkey

118. Talat PARVEEN Saudi Arabia

119. Manoj Kumar PATEL India

120. Rumi Melih PELEN Turkey

121. Stephen J. PRIDE UK

122. Edmund PUCZYLOWSKI Poland

123. Dilek PUSAT Turkey

124. Hourong QIN China

125. Nadeem Ur RAHMAN India

126. Udhayakumar RAMALINGAM India

127. Kulumani M. RANGASWAMY USA

128. P. G. ROMEO India

129. Mohammad SADEK Egypt

130. Ali M. SAGER Libya

131. Recep SAHIN Turkey

132. Serap SAHINKAYA Turkey

133. Burcin SALTIK Turkey

134. Bulent SARAC Turkey

135. Zehra SARIGEDIK Turkey

136. Umit SARP Turkey

137. Phill SCHULTZ Australia

138. Liang SHEN China

139. Faiza SHUJAT India

140. Patrick SMITH UK

141. David SSEVVIIRI South Africa

142. Ebru SOLAK Turkey

143. Gokhan SOYDAN Turkey

144. Lia VAS USA

145. Yahya TALEBI Iran

146. Adnan TERCAN Turkey

147. Rick THOMAS UK



148. Sultan Eylem TOKSOY Turkey

149. Ergun TURKMEN Turkey

150. Zubeyir TURKOGLU Turkey

151. Burcu UNGOR Turkey

152. Emel UNVER DEMIR Turkey

153. Hamid USEFI Canada

154. Indah Emilia WIJAYANTI Indonesia

155. Robert WISBAUER Germany

156. Melek YAGCI Turkey

157. Feyza YALCIN Turkey

158. Ramazan YASAR Turkey

159. Nihan BAYDAR YARBIL Turkey

160. Ece YETKIN Turkey

161. Ebru YIGIT Turkey

162. Filiz YILDIZ Turkey

163. Nazli YILDIZ IKIKARDES Turkey

164. Erol YILMAZ Turkey

165. Utku YILMAZTURK Turkey

166. Figen YUZBASI ERYILMAZ Turkey

167. Parvaneh ZOLFAGHARI Armenia


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