by ameer al-abayechi phd student, university of debrecen ...herfort/gtg_wien_17/aa.pdfΒ Β· let π:...
TRANSCRIPT
This presentation has two Subject
* Near Prime Spectrum
* Pre-Open Sets In Minimal Bitopological Spaces
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* Near Prime Spectrum
In algebraic geometry and commutative algebra, the Zariski
topology is a topology on algebraic varieties, introduced primarily by
Oscar Zariski a Russian-born American mathematician . And later
generalized for making the set of prime ideals of a commutative ring a
topological space, called the spectrum of the ring. 3
The generalization of the Zariski topology introduced by Hilbert
who suggested defining the Zariski topology on the set of the maximal
ideals of a commutative ring as the topology such that a set of
maximal ideals is closed if and only if it is the set of all maximal ideals
that contain a given ideal. Thus the Zariski topology on the set of
prime ideals (spectrum) of a commutative ring is the topology such
that a set of prime ideals is closed if and only if it is the set of all
prime ideals that contain a fixed ideal. 4
Introduction
Let πΉ be a ring with identity . The theory of the prime
spectrum of πΉ where πΊπππ(πΉ) = *π· βΆ π· ππ π πππππ ππ πππ ππ πΉ+ has
been developed since 1930 . The modern theory was developed by
Jacobson and Zariski mainly . The topology was defined on πΊπππ(πΉ) is
the collection of closed sets to be π½ π° = π· β πΊπππ πΉ : π° β π· it is
called the Zariski topology on πΊπππ(πΉ) . 5
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In this paper , we expanded Zariski
topology by introducing a new generalization of
near-ring using the near completely prime ideal
, called near Zariski topology .
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Definition :-
Let π be a nonempty set with two binary operations
(+) , (. ) . (π, +, . ) is called near-ring if and only if :
1. (π, +) is a group (not necessarily commutative) .
2. (π, . ) is a semi group .
3. For πππ π1, π2, π3 β π ; π1 + π2 . π3 = π1. π3
+ π2. π3(right distributive law) . This near-ring will be
termed as right near-ring .
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Remarks :-
β’ If π1. π2 + π3 = π1. π2 + π1. π3instead of condition (3)
the set N satisfies , then we call π a left near-ring .
β’ If 1. π = π (π. 1 = π) then π has a left identity(right
identity ) .
β’ If (π, +) is abelian , we call π an abelian near-ring .
β’ If (π, . ) is commutative we call π itself a commutative
near-ring . Clearly if π is commutative near-ring then left
and right distributive law is satisfied and 1. π = π. 1 = π ,
π is called unital commutative near-ring .
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Example :-
Near-ring arise very naturally in the study of
mappings on groups . If (πΊ, +) is a group (not
necessarily abelian ) then the set π(πΊ) of all
mappings from πΊ to πΊ is a near-ring with respect to
pointwies addition and composition of mappings .
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Definition :- Let π be a near-ring and πΌ is a nonempty subset of π . (πΌ, +, . ) is called an ideal of π if and only if : 1. (πΌ, +) is normal subgroup of (π, +) . 2. πΌπ β πΌπ and for all π, π1 β π and for all
π β πΌ , π π1 + π β ππ1 β πΌ . Definition :- An ideal π of π is said to be completely prime if ππ β π implies π β π or π β π for any π, π β π .
Let π be a commutative near-ring with identity and
π be a completely prime ideal of π . The set of all
completely prime ideal of π , is denoted by ππππ(π) is
called near prime spectrum on completely prime ideal . Let πΌ
is ideal of π and let π(πΌ) collection of all completely prime
ideal contains πΌ . The collection of all π(πΌ) satisfies the
axioms of closed sub sets of a topology for ππππ(π) called
the near Zariski topology for ππππ(π) . 11
Theorem 1 :- The near prime spectrum ππππ(π) of any commutative near-ring π is a compact topological space . Corollary 2 :- Let πΌ be a near ideal of near-ring π . Then the closed subset π(πΌ) of ππππ(π) is a compact set .
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Theorem 3 :-
Let π: π1 β π2 be a near-ring unital homomorphism between
near-rings π1 and π2 . Then π:π1 β π2 induces a continuous map
πβ: ππππ(π2) β ππππ(π1) , where πβ π = πβ1(π) for every
completely prime ideal π of π2 .
Proof :- Let π2 be a completely prime ideal of π2. Now 12 β π2,
because π2 is a proper ideal of π2, then 11 β πβ1 π2 , since
π 11 = 12. It follows that πβ1(π2) is a proper ideal of π1 . Let π₯ and
π¦ be elements of π1. Suppose that π₯π¦ β πβ1 π2 . Then π(π₯)π(π¦)
= π(π₯π¦) and therefore π π₯ π π¦ β π2. But π2 is a completely prime
ideal of π2, and therefore either π π₯ β π2 or π π¦ β π2 .Thus either
π₯ β πβ1 π2 or π¦ β πβ1 π2 . This shows that πβ1 π2 is a completely
prime ideal of π1.
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We conclude that there is a well-defined function
πβ: ππππ π2 β ππππ π1 such that πβ π2 = πβ1 π2 for all
completely prime ideal π2 of π2. Now we prove that πβ is a
continuous function , let πΌ1 be a ideal of π1 ,
πββ1 π πΌ1 = π2 β ππππ π2 : πβ π2 β π πΌ1 .
since πβ π2 = πβ1 π2
= π2 β ππππ π2 : πβ1 π2 β π πΌ1
= π2 β ππππ π2 : πΌ1 β πβ1 π2
= π2 β ππππ π2 : π πΌ1 β π2 = π π πΌ1
Thus , πβ: ππππ π2 β ππππ π1 is a continuous function .
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Theorem 4 :-
Let π be a near-ring , let πΌ be a proper ideal of π ,
and let ππΌ : π β π/πΌ be the corresponding quotient near-
ring homomorphism onto the quotient near-ring π/πΌ . Then
the induced map ππΌβ: ππππ π πΌ β ππππ (π) maps
ππππ π πΌ homeomorphically onto the closed set π(πΌ) .
Lemma 5 :-
If π is a noetherian near-ring . Then ππππ(π) is a
noetherian space .
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Lemma 6 :- Let π be a near-ring . Then the space ππππ(π) is a π0 space . Proof :- Let π1 , π2 β ππππ π and π1 β π2 then π1 β π2 or π2 β π1 , let π» π = π1 β ππππ π : π β π1 and π1 β π2. Then We get π1 β π» π , π2 β π» π . Thus ππππ π is a π0 space . Lemma 7 :- Let π be a near-ring . Then the space ππππ(π) is π1 if and only if ππππ(π) = πππ₯(π) is the set of all near maximal ideal of π . 16
* PRE-OPEN SETS IN MINIMAL BITOPOLOGICAL SPACES
17
Bitopological space started by Kelly 1968 is defined as
follows , a bitopological space is a set endowed with two
topologies. Typically, if the set is π and the topologies
are π and π then the bitopological space is referred to as
(π , π, π)
The notion of pre-open set was introduced by M. E. Abd
El-Monsef et al. 1982 .
The concept of minimal structure was introduced by V.
Popa and T. Noiri in 2000 18
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In this paper, we expanded bitopological by
introducing a new generalization, which uses a
minimal structure, called minimal bitopological
space . Then we studied pre-open sets in
minimal bitopological spaces with some results
and definitions of separation axioms on the
minimal bitopological .
Let π be a nonempty set and let π β π π we say that π
is minimal structure on π if β , π β π.
Let π be a non-empty set, π be a topology on π, let π be a
minimal structure on π then the triple (π, π,π) is called a
minimal bitopological space.
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Let (π, π, π) be a minimal bitopological space and π΄ be a
sub set of π, π΄ is called pre-open with respect to π and π
, if π΄ β πππ‘π(πππ(π΄)), where πππ(π΄) is the closure of π΄
with respect to π .
Let (π, π, π) be a minimal bitopological space , and π΄ β π
, the intersection of all pre-closed sets containing π΄ is
called pre-closure of π΄, and is denoted by ππππ β πΆπ A .
That is ππππ β πΆπ A is the smallest pre-closed set in π
containing π΄, also π΄ is pre-closed if and only if ππππ
β πΆπ A = π΄ .
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Theorem 1 :-
Let (π, π, π) be a minimal bitopological space, π β π,
π open with respect to π, π and let (π, ππ , ππ) be a sub
space of (π, π, π). If πΊ be pre-open in π with respect to π ,
π . Then πΊ β© π is pre-open in π with respect to ππ , ππ .
Corollary 2 :-
Let (π,π,π) be a minimal bitopological space, let
(π, ππ , ππ) be a sub space of (π, π, π) and let πopen with
respect to π , π . If πΊ be pre-closed in π with respect to π ,
π . Then πΊβ©π is pre-closed in π with respect to ππ , ππ . 23
Theorem 3 :-
A minimal bitopological space (π, π, π) is ππ β ππ-space if
and only if ππππ -closure of distinct points are distinct .
Proof :- Let π₯, π¦ β π, π₯ β π¦ implies ππππ β πΆπ π₯ β πππ
π β πΆπ(*π¦+).
Since ππππ β πΆπ π₯ β πππ
π β πΆπ(*π¦+) there exists at least one point π§,
such that π§ β ππππ β πΆπ π₯ , but π§ β πππ
π β πΆπ π¦ . We claim that
π₯ β ππππ β πΆπ π¦ . For, let π₯ β πππ
π β πΆπ π¦ . Then ππππ β πΆπ π₯
β ππππ β πΆπ(*π¦+) which is a contradiction that π₯ β πππ
π β πΆπ π¦ .
Hence π₯ β π\ππππ β πΆπ π¦ but πππ
π β πΆπ π¦ is πππ-closed, so
π\ππππ β πΆπ π¦ is πππ-open which contains π₯ but not π¦. It follows
that (π, π, π) is ππ β ππ-space.
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Conversely,
since (π, π, π) is ππ β ππ-space, then for each two
distinct points π₯, π¦ β π, π₯ β π¦ there exists πππ-open set πΊ
such that π₯ β πΊ, π¦ β πΊ . π\πΊ is πππ-closed set which does
not contain π₯ but contains π¦. By Definition , ππππ β πΆπ(*π¦+)
is the intersection of all πππ-closed sets which contain *π¦+.
Thus , ππππ β πΆπ(*π¦+) β π\πΊ, then π₯ β π\πΊ. This implies
that π₯ β ππππ β πΆπ(*π¦+) , but π₯ β πππ
π β πΆπ(*π₯+) , π₯ β ππππ
β πΆπ(*π¦+). Therefore ππππ β πΆπ π₯ β πππ
π β πΆπ π¦ .
Theorem 4 :- Every subspace of ππ β ππ-
space is ππ β ππ-space .
Theorem 5 :- Every subspace of ππ β π1-
space is ππ β π1-space .
Theorem 6 :- Every subspace of ππ β π2-
space is ππ β π2-space .
Theorem 7 :- Every subspace of a πππ-
regular space is a πππ-regular space.
Theorem 8 :- Every πππ-closed subspace
of a πππ-normal space is a πππ-normal.
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Theorem 9 :-
A minimal bitopological space (π, π, π) is an ππ β π1-
space If and only if every singleton subset *π₯+ of is πππ-
closed set.
Proof :- Suppose π is an ππ β π1-space. Then for π₯, π¦ β π
and π₯ β π¦, there are two πππ-open sets πΊ and π» such that
π₯ β π», π¦ β π» and π¦ β πΊ and π₯ β πΊ. So πΊ β *π₯+π also
βͺ *πΊ βΆ π¦ β π₯+ β *π₯+π and *π₯+πββͺ *πΊ: π¦ β π₯+ . Hence
*π₯+π=βͺ *πΊ: π¦ β π₯+, which is a πππ-open set. Then *π₯+ is a
πππ-closed . Conversely, Let π₯, π¦ β π and π₯ β π¦. If *π₯+ and
*π¦+ are the πππ-closed sets of π₯ and π¦ respectively such that
π₯ β *π¦+, then *π₯+π and *π¦+π are πππ-open sets such that
π¦ β *π₯+πand π₯ β *π₯+πalso π₯ β *π¦+πand π¦ β *π¦+π. Then π is
an ππ β π1-space . 27
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