literature on geometry - uni-due.de · 91 handbook of incidence geometry. in preparation, to be...

Journal of Geometry Vol. 42 (1991)

0047-2468/91/020180-21$1.50+0.20/0 (C) 1991. Birkhäuser Verlag, Basel

LITERATURE ON GEOMETRY OVER RINGS

Günter Törner and Ferdinand D. Veldkamp

The following is intended as a complete list of the literature on projective and affine geometry over rings which has appeared up to now. It includes the material froni two earlier sucli lists, viz.,

B. Artmann, G. Dorn, D. Drake, G. Törner: Hjelmslev'sche Inzidenzgeometrie und verwandte Gebiete - Literaturverzeichnis. J. Geom. 7 (1976), 175-191.

D. Jungnickel: Hjelmslev'sche Inzidenzgeometrie und verwandte Gebiete - Literaturver- zeichnis 11. J. Geom. 16 (1981), 138-147.

Articles on reflection geometry in the sense of F. Bachmann's book "Aufbau der Geometrie aus dem Spiegelungsbegriff", and on chain geometries in the sense of W. Benz' "Georrietrie der Algebren" are not included.

Papers on geometric algebra over rings (structure and automorphisms of classical groups over rings) have also been omitted with sonie exceptions; for systematic listings of sucli papers we refer to McDonald 76 and Hahn; O'Meara 89.

The authors will be grateful for additions, completions and corrections to t,his list.

ANDRE, J. :

69 Über Homomorphismen projektiver Ebenen. Abh. Math. Sem. b'niti. Hnmbury 34 (1969), 98-114.

72 Über verallgemeinerte Traiislationsstrukturen und Homomorphismen. Arch. Malh. (Base l ) 23 (1972), 198-205.

ARCHBOLD, J.W. : 55 Projective geornetry over an algcbra. Mnthemntika 2 (1955), 105-115.

Torner and Veldkamp

ARGUNOV, B.I.; EMEL'CHENKOV, E.P. : 82 Incidence structures and ternary algebras. Russian Math. Surveys 37 : 2 (1982), 1-39.

ARMENTROUT, N.; HARDY, F.L.; MAXSON, C.J. : 74 On generalized affine planes. J. Geom. 4 (1974), 143-159.

ARNOLD, H.J. : 71a A way t o the geometry of rings. J. Geom. 1 (1971), 155-168.

71b Die Geometrie der Ringe im Rahmen allgemeiner affiner Strukturen. Hamburger Math. Einzelschriften, Neue Folge 4 (1971).

ARTMANN, B. : 69a Hjelmslev planes derived from modular lattices. Canad. J. Math. 21 (1969), 76-83.

69b Hjelmslev-Ebenen mit verfeinerten Nachbarschaftsrelationen. Math. Z. 112 (1969), 163- 180.

69c Uniforme Hjelmslev-Ebenen und modulare Verbände. Math. Z. 111 (1969), 15-45.

70a Hjelmslev-Ebenen in projektiven Räumen. Arch. Math. (Basel) 21 (1970), 304-307.

70b Über die Einbettung uniformer affiner Hjelmslev-Ebenen in projektive Hjelmslev-Ebenen. Abh. Math. Sem. Univ. Hamburg 34 (1970), 127-134.

71a Existenz und projektive Limiten von Hjelmslev-Ebenen n-ter Stufe. In: Atti Conv. "Geo- metria Combinatoria" (Perugia), 1970, pp. 27-41. Perugia, 1971.

71b Desarguessche Hjelmslev-Ebenen n-ter Stufe. Mitt. Math. Sem. Giefien 91 (1971), 1-19.

72 Geometric aspects of primary lattices. Pacific J. Math. 43 (1972), 15-25.

BACON, P.Y. : 71 On Hjelmslev planes with small invariants. Masters Thesis, Univ. of Florida, Gainesville,

FL, 1971. 72 Strongly n-uniform and level n Hjelmslev planes. Math. 2. 127 (1972), 1-9.

74a Coordinatized Hjelmslev planes. Diss. Univ. of Florida, Gainesville, FL, 1974. 74b On the extension of projectively uniform affine Hjelmslev-planes. Abh. Math. Sem. Univ.

Hamburg 41 (1974), 185-189.

76 An introduction to Klingenberg planes 1. (232 pp.) Published by P.Y.Bacon, 3101 NW 2nd Av., Gainesville, FL 32607, 1976.

78 Desarguesian Klingenberg planes. Trans. Amer. Math. Soc. 241 (1978), 343-355.

79 An introduction to Klingenberg planes 2 and 3. (317 resp. 402 pp). Published by P.Y. Bacon, 3101 NW 2nd Av., Gainesville, FL 32607, 1979.

BAER, R. : 42 A unified theory of projective spaces and finite abelian groups. Trans. Amer. Math. Soc.

52 (1942), 283-343.

62 Hjelmslevsche Geometrie. In: Algebraical and Topological Poundations of Geometry (Proc. Coll. Utrecht, 1959), pp. 1-4. Pergamon, Oxford, 1962.

68 1) Kollineationen primärer Praemoduln. 2) Dualisierbare Moduln und Praemoduln. In: Studies on Abelian Groups (Symposium, Montpellier, 1967), pp. 1-36 and 37-67. Springer- Verlag, Berlin, 1968.

182 Törner and Veldkamp

BAKER, C.A. : 78a Moulton affine Hjelmslev planes. Canad. Math. Bull. 21 (1978), 135-142.

78b Ordered Hjelmslev planes. Ph. D. Thesis, McMaster Univ., Hamilton, Ont., 1978. 84 Ordered affine Hjelmslev planes. J. Geom. 23 (1984),1-13.

85 Preordered uniform Hjelmslev planes. J. Geom. 24 (1985), 14-17.

BAKER, C.A.; LANE, N.D.; LAXTON, J.A.; LORIMER, J.W. : 84 Preordered affine Hjelmslev planes. J. Geom. 23 (1984), 14-44

BAKER, C.A.; LANE, N.D.; LORIMER, J.W. : 82 Order and topology in projective Hjelmslev planes. J. Geom. 19 (1982), 8-42.

84 Preordered affine Hjelmslev planes. J. Geom. 23 (1984), 8-42.

88a A construction for topological non-Desarguesian affine Hjelmslev planes. Arch. Math. (Basel) 50 (1988), 83-92.

88b Local alternative rings and Moufang projective Klingenberg planes. Math. Report 127 (1988), McMaster Univ., Hamilton, Ont.

90a A coordinatization for Moufang Klingenberg planes. To appear in Simon Stevin

90b Local alternative rings and finite alternative chain rings. C. R. Math. Rep. Acad. Sci. Canada 12 (1990), no.2, 53-58.

90c The Artin-Zorn theorem for finite punctually cohesive projective Klingenberg planes. Ars. Combin. 29 B (1990), 143-149.

90d An affine charactenzation of Moufang projective Klingenberg planes. Resultate Math. 17 (1990), 27-36.

BARBILIAN, D. : 40 Zur Axiomatik der projektiven ebenen Ringgeometrien I. Jahresber. Deutsch. Math.- Verein

50 (1940), 197 - 229.

41 Zur Axiomatik der projektiven ebenen Ringgeometrien 11. Jahresber. Deutsch. Math.- Verein. 51 (1941), 34 - 76.

BARTOLONE, C.; BARTOLOZZI, F. : 85 Topics in geometrk algebra over rings. In: Rings and geometry ( N A T O Adv. Study Inst.,

Istanbul, 1984), pp. 353-389. Reidel, Dordrecht, 1985.

BARTOLONE, C.; D1 FRANCO, F. : 79 A remark on the projectivities of the projective line over a commutative ring. Math. Z.

169 (1979), 23-29.

BENZ, W. : 63 Süßsche Gruppen in affinen Ebenen mit Nachbarelementen und allgemeineren Strukturen.

Abh. Math. Sem. Univ. Hamburg 26 (1963), 83-101.

66 0-Geometrie und Geometrie von Hjelmslev. Math. Ann. 164 (1966), 118-123.

74 Ebene Geometrie über einem Ring. Math. Nachr. 59 (1974), 163-193.

79 On Barbilian domains over commutative rings. J. Geom. 12 (1979), 146- 151.

Törner and Veldkamp 183

85 On mappings preserving a single Lorentz-Minkowski distance in Hjelmslev's classical plane. Aequationes Math. 28 (1985), 80-87.

BETSCH, CH. : 17 Zur analytischen Geometrie der dualen Größen. Diss. Tübingen, 1917 (62 p.)

BILO, J.; DEPUNT, J. : 62 On the linear transformations of the ternion line. Med. Konink. Acad. Wetensch. Belgie,

1962, nr. 8. (Dutch).

BINGEN, F. : 66 Geometrie projective sur un anneau semiprimaire. Acad. Roy. Belg. Bull. Cl. Sci. (5) 5 2

(1966), 13-24.

BISZTRICZKY, T.; LORIMER, J.W. : 90 Axiom systems for affine Klingenberg spaces. In: Proc. Conf. Combinatorics '88 (Rauello,

1988), to appear.

9 1 On hyperplanes and free subspaces of affine Klingenberg spaces. Preprint, Univ. of Toronto (1991).

BIX, R. : 80 Octonion planes over local rings. Trans. Amer. Math. Soc. 261 (1980), 417-438.

BOTTO-MURA, R.I.; BRUNGS, H.H.; FISHER, J.L. : 77 Chain rings and valuation semigroups. Comm. Algebra 5 (1977), 1529-1547

BREHM, U. :

83 Untermodulverbände torsionsfreier Moduln. Diss. Univ. Freiburg im Br., 1983. 85 Coordinatization of lattices. In: Rings and geometry (NATO Adu. Study Inst., Istanbul,

1984), pp. 511-550, Reidel, Dordrecht, 1985.

87 Representations of homomorphisms between submodule lattices. Resultate Math. 12 (1987), 62-70.

BRUNGS, H.H.; TÖRNER, G. : 76 Chain rings and prime ideals. Arch. Math. (Basel) 27 (1976), 253-260.

78 Embedding right chain rings into chain rings. Canad. J. Math. 30 (1978), 1079-1086.

BUEKENHOUT, F. (editor) : 91 Handbook of incidence geometry. In preparation, t o be published by North Holland publ.

comp.

BURIAN, K. : 72 Affine H-structures. Comment. Math. Univ. Carolin. 13 (1972), 629-635.

74 Affine parallel H-structures. Sb. Praci Ped. Fak. V Ostraue Ser. A 9 (1974), no 37, 3-5. (Czech.)

75 On finite Hjelmslev nets. Knitnice Odborn B 56 (1975), 69-74. (Czech.)

CAMERON, P.J.; DRAKE, D.A. : 80 Partial X-geometries of small nexus. Ann. Discr. Math. 6 (1980), 19-29.


CARTER, D.S.; VOGT, A. : 80 Collinearity preserving functions between Desarguesian planes. Mem. Amer. Math. Soc.

27, No. 235. Providence, R.I., 1980.

CIVOLANI, N. : 80 Hyperfree extensions of partial Klingenberg planes. Geom. Dedicata 9 (1980), 467-475.

CLARK, W.E.; DRAKE, D.A. : 73 Finite chain rings. Abh. Math. Sem. Univ. Hamburg 39 (1973), 147-153.

CLARK, W.E.; LIANG, J . : 73 Enumeration of finite commutative chain rings. J. Algebm 27 (1973), 445-453.

CRAIG, R.T. : 64 Extensions of finite projective planes. 1-uniform Hjelmslev planes. Canad. J. Math. 16

(1964), 261-266.

CRONHEIM, A. : 76 Cartesian groups, formal power series and Hjelmslev planes. Arch. Math. (Basel) 27

(1976), 209-220.

78 Dual numbers, Witt vectors and Hjelmslev planes. Geom. Dedicata 7 (1978), 287-302.

CYGANOVA, V.K. : See TSYGANOVA, V.K. :

DEMBOWSKI, P. : 68 Fkt<GeÖmeGies.-~@b. Matk 444. Springer-Verlag Berlin-1968. - - - - -

DEPUNT, J . : 60 Foundations of analytic projective ternion geometry of the plane. Verh. Konink. Acad.

Wetensch. Belgie 63 (1960). (Dutch)

DE WINNE, P. : 74 An extension of the notion "cross-ratio of an ordered 4-tuple of points of the projectivc

line" to an ordered (n + 3)-tuple of points (resp. hyperplanes) of the n-dimensional space over an arbitrary ring with identity, I and 11. Simon Stevin 47 (1974), 139-159 and 48 (1974), 41-56.

DORN, G. : 74 Affine Geometrie über Matrizenringen. Mitt. Math. Sem. GieJlen 109 (1974), 1-120.

DRAKE, D.A. : 67 Neighbor collineations and affine-projective extensions of Hjelmslev planes. Diss. Syracuse

Univ. Syracuse, N. Y., 1967. (110 p.)

68 Projective extensions of uniform affine Hjelmslev planes. Math. 2. 105 (1968), 196-207.

70a Coordinatization of H-planes by H-modules. Math. 2. 115 (1970), 79-103.

70b On n-uniform Hjelmslev planes. J. Combin. Theory 9 (1970) , 267 - 288.

71 The translation groups of n-uniform translation Hjelmslev planes. Pacific J. Math. 38 (1971), 365-376. - - - - - -

----P- ----


The structure of n-uniform translation Hjelmslev planes. Trans. Amer. Math. Soc. 175 (1973), 249-282.

Near affine Hjelmslev planes. J. Combin. Theory Ser. A 16 (1974), 34-50.

Nonexistence results for finite Hjelmslev planes. Abh. Math. Sem. Hamburg 40 (1974), 100-110. Existente of parallelisms and projective extensions for strongly n-uniform near affine Hjelmslev planes. Georn. Dedicata 3 (1974), 191-214.

Affine Hjelmslev-Ebenen mit verfeinerten Nachbarschaften. Math. 2. 143 (1975), 15-26.

All n-uniform quasitranslation Hjelmslev planes are strongly n-uniform. Proc. Amer. Math. Soc. 51 (1975), 494-498.

More new integer pairs for finite Hjelmslev planes. nlinois J. Math. 19 (1975), 618-627.

Nonexistence results for finite Hjelmslev planes 11. Preprint, 1975. Charakterisierungen der Hjelmslev-Ebenen mit Invarianten (4,2). Arch. Math. (Basel) 27 (1976), 436-440.

Construction of Hjelmslev planes. J. Geom. 10 (1977), 183-197.

Partial X-geometries and generalized Hadamard matrices over groups. Canad. J. Math. 31 (1979), 617-627.

The use of awiliary sets of matrices in the construction of Hjelmslev and Klingenberg structures. In: Finite geometries (Pullman, Wash., 1981) pp. 129-153. Lecture Notes in Pure and Appl. Math. 82. Dekker, New York, 1983. Squeezing the accordion in a strongly n-uniform Hjelmslev plane. Math. 2. 185 (1984), 151-166.

DRAKE, D.A.; FREEMAN, J.W. : 79 Partial t-spreads and group constructible (s,r,p)-nets. J. Geom. 13 (1979), 210-216.

DRAKE, D.A.; HALE, M.P. : 77 Group constructible (t, k)-nets and Hjelmslev planes. J. Algebm 48 (1977), 321-331.

DRAKE, D.A.; JUNGNICKEL, D. : 77 Klingenberg structures and partial designs I: Congruence relations and solutions. J .

Statist. Plann. Inference 1 (1977), 265-287.

78 Klingenberg structures and partial designs 11. Regularity and uniformity. Pacific J. Math. 77 (1978), 389-415.

80 Das Existenzproblem für projektive (8,5)-Hjelmslevebenen. Abh. Math. Sem. Univ. Ham- burg 50 (1980), 118-126.

85 Finite Hjelmslev planes and Klingenberg epimorphisms. In: Rings und geometry (NATO Adv. Study Inst., Istanbul, 1984), pp. 153-231. Reidel, Dordrecht, 1985.

DRAKE, D.A.; LENZ, H. : 75 Finite Klingenberg planes. Abh. Math. Sem. Univ. Hamburg 44 (1975), 70-83.

DRAKE, D.A.; SANE, S.S. : 82a Auxiliary sets of matrices with new step parameter sequences. Linear Algebra Appl. 46

(1982), 131-153.


82b Maximal intersecting families of finite sets and n-uniform Hjelmslev planes. Proc. Amer. Math. Soc. 86 (1982), 358-362.

DRAKE, D.A.; SHULT, E.E. : 76 Construction of Hjelmslev planes from (t, k)-nets. Geom. Dedicata 5 (1976), 377-392.

DRAKE, D.A.; TÖRNER, G. : 76 Die Invarianten einer Klasse von projektiven Hjelmslev-Ebenen. J. Geom. 7 (1976), 157-

174.

DUGAS, M. : 74 Charakterisierungen endlicher desarguesscher uniformer Hjelmslev-Ebenen. Geom. Dedi-

cata 3 (1974), 295-324.

76 Eine Kennzeichnung der endlichen desarguesschen Hjelmslev-Ebenen. Arch. Math. (Basel) 27 (1976), 556-560.

77 Moufang-Hjelmslev-Ebenen. Arch. Math. (Basel) 28 (1977), 318-322.

79 VeraUgemeinerte Andre-Ebenen mit Epimorphismen auf Hjelmslev-Ebenen. Geom. Dedi- cata 8 (1979), 105-123.

80a Der Zusammenhang zwischen Hjelmslev-Ebenen und H-Verbänden. Mitt. Math. Ges. (Hamburg) 10, no. 8, (1980), 709-749.

80b Die Koiiineationsgruppe verallgemeinerter Andre-Ebenen mit Homologien. Math. Z. 175 (1980), 87-95.

ELISEEV, E.M. :

75 Pappus' theorem in Hjelmslev projective space. Smolensk. Gos. Ped. Inst. 1975, deposited at Viniti 30-76. (Russian)

EMEL'CHENKOV, E.P. : Ternars and automorphisms of Hjelmslev planes. Ph. D. thesis Smolensk, 1972. (Russian)

Translation AH-planes and H-ternaries. Smolenk. Gos. Ped. Inst. Uchen. Zap. 4 (1973), 7483. (Russian)

The PH-ternary of a Hjelmslev projective plane. Smolenk. Gos. Ped. Inst. Uchen. Zap. 4 (1973), 93-101. (Russian)

Displacements of an affine Hjelmslev plane. Smolensk. Gos. Ped. Inst. 1974, deposited at Viniti 2162-74. (Russian)

Projective reflections of H-planes and RH-ternary rings. Smolensk 1974, 8 pp. (Russian)

Affine Translation H-planes. Smolensk 1974, 11 pp. (Russian)

Homotheties of AH-planes and H-Ternaries. Leningmd. Gos. Ped. Inst., Geometry und Topology No. 2, (1974), 89-93. (Russian)

On biternary rings and automorphisms of Hjelmslev planes. Moscow, 1975. (Russian)

(n,l)-transitive AH-planes. Smolensk. Gos. Ped. Inst. 1977, deposited at Viniti 1540-78. (Russian)

(II,I)-transitive non-Fano AH-planes and H-ternars. Smolensk. Gos. Ped. Inst. 1977, deposited at Viniti 1977-77. (Russian)

1 Törner and Veldkamp 187

~ 78 (II,l)-collineations of AH-planes. Modern Geometry (Russian), pp. 58-60. Leningrad.

I Gos. Ped. Inst., Leningrad, 1978. (Russian)

EMEL'CHENKOV, E.P.; KUR'YAKOV, M.M. : 78 Affine Hjelmslev pseudoplanes and AHP-ternars. Smolensk. Gos. Ped. Inst. 1978, de-

posited at Viniti 3905- 178. (Russian)

1 EMELTSCHENKOV, E.P.: see EMEL'CHENKOV, E.P. :

i EVERETT, C.J. : 42a Vector spaces over rings. Bull. Amer. Math. Soc. 48 (1942), 312-316.

42b Affine geometry of vector spaces over rings. Duke Math. J. 9 (1942), 873-878.

45 The basis theorem of vector spaces over rings. Bull. Amer. Math. Soc. 5 1 (1945), 531-532.

FAIGLE, U.; HERRMANN, C. : 81 Projective geometry on partially ordered sets. Tmns. Amer. Math. Soc. 266 (1981),

319-332.

FAULKNER, J.R. : 70 Octonion planes defined by quadratic Jordan algebras. Mem. Amer. Math. Soc., Nr. 104.

Providence, R.I., 1970.

83a Stable range and linear groups for alternative rings. Geom. Dedicata 14 (1983), 177-188.

83b Coordinatization of Moufang-Veldkamp planes. Geom. Dedicata 14 (1983), 189-201.

89a A geometric construction of Moufang planes. Geom. Dedicata 2 9 (1989), 133-140.

89b Barbilian planes. Geom. Dedicata 30 (1989), 125-181.

FAULKNER, J.R.; FERRAR, J.C. : 85 Generalizing the Moufang plane. In: Rings and geometry (NATO Adv. Study Inst., Istan-

bul, 1984), pp. 235-288. Reidel, Dordrecht, 1985.

FERRAR, J.C.; VELDKAMP, F.D. : 85 Neighbor-preserving homomorphisms between projective ring planes. Geom. Dedicata 18

(1985), 11-33.

87 Admissible subrings revisited. Geom. Dedicata 23 (1987), 229-236.

FRITSCH, R.; PRESTEL, A. : 82 Bewertungsfortsetzungen und nicht-injektive Kollineationen. Geom. Dedicata 13 (1982),

107-111.

GARNER, L.H. : 72 Fields and projective planes: a category equivalence. Rocky Mountain J. Math. 2 (1972),

605-610.

GEGENWART, M. :

87 Gruppentheoretische Darstellung der fanoschen affinen Hjelmslev-Ebenen über lokalen Ringen. Diss. TH-Darmstadt (FRG), 1987.

GEVAERT, H.; KEPPENS, D. : 87 On ovals in finite 2-uniform projective Hjelmslev planes. Preprint Univ. of Ghent, 1987.


GIBERT, K. YA. :

83 Projective spaces over algebras and C,(R, A)-geometries. Geom. of imbedded manifolds, 33-37, 121. Moskov. Gos. Ped. Inst., Moscow, 1983. (Russian)

GROZE, V.; VASIU, A. : 80 Affine structures over an arbitrary ring. Studia Univ. Babeg-Bolyai Math. 25 (1980),

28-31.

G R U N D H ~ F E R , T. : 88 Compact disconnected planes, inverse limits and homomorphisms. Monatsh. Math. 105

(1988), 261-277.

G R ~ N W A L D , J. : 06 Über duale Zahlen und ihre Anwendung in der Geometrie. Monatsh. Math. und Physik

17 (1906), 81-136.

HAHN, A.J.; O'MEARA, O.T. : 89 The classical groups and K-theory. Crvndlehren der math. Wissensch. 291. Springer-

Verlag, Berlin 1989.

HALE, M.P.; JUNGNICKEL, D. : 78 A generalization of Singer's theorem. Proc. Amer. Math. Soc. 71 (1978), 280-284

HANSSENS, G.; VAN MALDEGHEM, H. : 89a A note on near-Barbilian planes. Geom. Dedicata 29 (1989), 233-235.

89b On projective Hjelmslev planes of level n. Glasgow Math. J . 31 (1989), 257-261.

89c A universal construction for projective Hjelmslev planes of level n. Compositio Math. 71 (1989), 285-294.

HJELMSLEV, J. : Die Geometrie der Wirklichkeit. Acta Math. 40 (1916), 33-66.

Die natürliche Geometrie. Abh. Math. Sem. Univ. Hamburg 2 (1923), 1-36. Einleitung in die allgemeine Kongruenzlehre I. Danske Vid. Selsk. Mal.-Fys. Medd. 8 nr. 11 (1929).

Einleitung in die allgemeine Kongruenzlehre 11. Danske Vid. Selsk. Mat.-Fys. Medd. 10 nr. 1 (1929).

La g6om6tne sensible I. Enseignernent Math. 38 (1940), 7 - 27.

Einleitung in die allgemeine Kongruenzlehre 111. Danske Vid. Selsk. Mal.-Fys. Medd. 19 nr. 12 (1942).

La g6ometrie sensible 11. Enseignement Math. 38 (1942), 294 - 322.

La ghmetrie sensible 111. Enseignement Math. 39 (1942-1950), 210 - 236.

Einleitung in die allgemeine Kongruenzlehre IV. Danske Vid. Selsk. Mat.-Fys. Medd. 22 nr. 6 (1945).

Einleitung in die allgemeine Kongruenzlehre V. Danske Vid. Selsk. Md.-Fys. Medd. 22 nr. 13 (1945).


49 Einleitung in die allgemeine Kongruenzlehre VI. Danske Vid. Selsk. Mat.-Fys. Medd. 25 nr. 10 (1949).

HUBAUT, X. : 64 Construction d'une droite projective sur Une algebre associative. Acad. Roy. Belg. Bull.

Cr. Sci. ( 5 ) 50 (1964), 618-623.

INABA, E. : 48 On primary lattices. J. Fac. Sci. Hokkaido Univ. Ser. I, 11 (1948) , 39-107.

51 Some remarks on primary lattices. Nat. Sci. Rep. Ochanomizu Univ. 3 (1951) , 1-5.

JAMES, D.G. :

73 On the structure of ortogonal groups over local rings. Amer. J. Math. 95 (1973) , 255-265.

80 Projective geometry for orthogonal groups. J. Reine Angew. Math. 319 (1980) , 104- 11 7 .

88 Projective geometry over rings with stable range condition. Linear and Multilinear Algebra 23 (1988) , 299-304.

JONSSON, B.; MONK, G.S. : 69 Representations of pnmary arguesian lattices. Pacific J. Math. 30 (1969) , 95-139.

JUNGNICKEL, D. : Klingenberg and Hjelmslev planes. Master's thesis, Freie Uniu., Berlin, 1975. Verallgemeinerte Klingenberg-Ebenen. Mitt. Math. Sem. GiePen 120 (1976) , 1-10.

Affine TD-Ebenen. Mitt. Math. Sem. GieJen 120 (1976) , 39-60.

Konstruktion transitiver Inzidenzstrukturen mit Differenzenverfahren. Diss., Freie Univ. Berlin, 1976.

Hjelmslev-Ebenen mit regulärer abelscher Kollineationsgruppe. In: Beiträge zur geometrischen Algebm (Duisburg, 1976), pp. 157-165. Birkhäuser, Basel, 1977.

On some classes of finite Hjelmslev planes. In: Proc. 8th. Southeastern Conf. on Combi- natorics, Graph Theory and Computing (Ba ton Rouge, LA, 1977), pp. 423-432. Utilitas Math. Publ. Comp., Winnipeg, 1977.

Reguläre Klingenberg-Strukturen und Hjelmslev-Ebenen. Habilitationsschrift, Freie Univ. Berlin, 1977. On the congruence relations of regular Klingenberg structures. J. Combin. Inform. System Sci. 3 (1978), 49-57.

On TD-structures. Arch. Math. (Basel) 31 (1978) , 403-413.

Regular Hjelmslev planes 11. Tmns . Amer. Math. Soc. 241 (1978), 321-330.

Construction of regular proper CK-planes. J. Combin. Inform. Sys tem Sci. 4 (1979), 14-18. Die Methode der Hilfsmatrizen. In: Contrib. Geom. (Siegen, 1978), pp. 388.394. Birkhäuser, Basel, 1979.

On an assertion of Dembowski. J. Geom. 12 (1979) , 168-174.

On balanced regular Hjelmslev planes. Geom. Dedicata 8 (1979) , 445-462.

Törner and Veldkamp

Regular Hjeimslev planes. J. Combin. Theory Ser. A 26 (1979), 20-37.

Some new combinatorid results on finite Klingenberg structures. Utilitas Math. 16 (1979), 249-269. A class of uniform Klingenberg matrices. Ars Combin. 10 (1980), 91-94.

Existence results for translation nets. In: Finite Ceometries and Designs (Chelwood Gate, 1980), pp. 172-196. London Math. Soc. Lecture Note Ser. 49. Carnbridge Univ. Press, 1981. Existence results for translation nets 11. J. Algebra 122 (1989), 288-298.

A family of relative difference sets associated with Hjelmslev structures over finite projective spaces. J. Geom. 35 (1989), 103-107.

KAPRALOVA, S.B. :

80 Dual Galois spaces. Zkidu Geom. Sem. Kazan. Univ. No. 12 (1980), 38-44. (Russian)

KERAWALA, S.M. : 41 Euclidean Aspect of Hjelmslev Geometry. P m . oj the Benares Math. Soc. N.S. 3 (1941)

39-53.

KEPPENS, D. : Möbius planes with neighbor relation. Simon Stevin 61 (1987), 157-170

Laguerre and Minkowski planes with neighbor reiation. J. Geom. 30 (1987), 12-27. Polanties in finite 2-uniform projective Hjelmslev Planes. Geom. Dedicata 24 (1987), 51-76. On polarities in the k-uniform n-dimensional projective Hjelmslev space PH(n, G ~ ( q ) [ t ] / t ~ ) q odd. Resultate Math. 12 (1987), 297-324. Klingenberg incidence structures. Diss. Univ. o j Ghent, 1987. (Dutch)

Polanties in n-uniform projective Hjelmslev planes. Geom. Dedicata 26 (1988), 185-214.

Coordinatization of projective Klingenberg planes, I and 11. Simon Stevin 62 (1988), 63-90 and 163-188. Neighbor-preserving epimorphisms between projective Klingenberg planes. Geom. Dedi- cata 29 (1989), 209-219.

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77a The fundamental theorem for the projective line over commutative rings. Aequationes Math. 16 (1977), 275-281.

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70 Projektive Hjelmslevräume. J. Reine Angew. Math. 243 (1970), 121-158.

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Homomorphismen von projektiven Räumen und verallgemeinerte semilineare Abbildun- gen. Chsopis Pest. Mat. 100 (1975), 142-154. Desarguessche affine Ebenen mit Homomorphismus. Geom. Dedicata 3 (1975), 493-509.

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Biternärringe und affine lokale Ternärringe. Acta Uniu. Palack. Olumuc., Fac. Reruna Natur. Math. 27 (1988), 25-37. Fastgeordnete affine lokale Ternärringe und pregeordnete Biternärringe. Actu Univ. Palack. Olumuc., Fac. Rerum Natur. Math. 28 (1989), 11-25.


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Homomorphismen projektiver Räume und Hjelmslevsche Geometrie. J. Reine Angew. Math. 254 (1972), 41-73.

Ein Beweis der Dimensionsformel in projektiven Hjelmslevschen Räumen. J. Reine Angew. Math. 256 (1972), 215-220.

Kennzeichnende Eigenschaften bewerteter Vektorräume. J. Reine Angew. Math. 260 (1973), 127-132.

I-Hüile und I-Kern von Moduln über Bewertungsringen. J. Reine Angew. Math. 2831284 (1976), 1-8.

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Die Bedeutungen nicht-invarianter Bewertungen für Hjelmslevsche Geometrien. In: Beiträge zur geometrischen Algebm (Duisburg, 1976), pp. 259-263. Birkhäuser, Basel, 1977. Projektive Hjelmslevsche Räume im nicht-invarianten Fall. J. Reine Angew. Math. 291 (1977), 182-188.

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Dekker, New York, 1976. 81 Projectivities over rings with many units. Comm. Algebra 9 (1981), 195-204.

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An. Stiinf. Univ. "Al. I. Cuza" lasi Secf. I a Mut. (N .S . ) 24 (1978), 15-19.

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70 Non-injective collineations on some sets in Desarguesian projective planes and extension of non-commutative valuations. Aequationes Math. 4 (1970), 307-321.

77 A generalization of the valued vector space. Anal. Numer. Theor. Approz. 19 (42) (1977), 203-209.

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81a New integer pairs for Hjelmslev planes. Geom. Dedicata 10 (1981), 35-48.

81b Some new invariant pairs ( t ,3) for projective Hjelmslev planes. J. Geom. 15 (1981), 64-73.

81c On class regular Hjelmslev planes. In: Finite Geometries and Designs (Chelwood Gate, 1980), pp. 332-336. London Math. Soc. Lecture Note Ser. 49. Cambridge Univ. Press, 1981.

81d On the theorems of Drake and Lenz. Aequationes Math. 23 (1981), 223-232.

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THAS, J.A. : Cross ratio of four points on the projective line over an associative algebra with identity. Simon Steuin 42 (1969), 97-111. (Dutch)

A study of the projective line over the total matrix algebra M3(Ii) of 3 X 3 matrices with entries in an algebraically closed field. Verh. Konink. Acad. Wetensch. Belgie 31, Nr. 112 (1969). (Dutch)

The m-dimensional projective space S,(M,,(GF(q))) over the total matrix aigebra M,(GF(q)) of the n X n-matrices with elements in the Gaiois field GF(q). Rend. Mat. 4 (1971), 459-532.

Deduction of properties, valid in the projective space S3,-,(K) using the projective plane over the total matrix algebra M,(K). Simon Steuin 46 (1972), 3-16.

4-gonal condigurations. In: Finite geometric structures and their applications ( C I . M. E. Bressanone, 1972), pp. 249-263. Edizioni Cremonese, Roma, 1973.

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78 Ordered desarguesian affine Hjelmslev planes. Canad. Math. Bull. 21 (1978), 229-235.

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76a Bemerkungen zu (6,2)-Hjelmslev-Ebenen. Mitt. Math. Sem. GieJen 121 (1976), 93-101.

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77c Über den Stufenaufbau von Hjelmslev-Ebenen. Mitt. Math. Sem. GiePen 126 (1977).

77d Über ein Problem von Klingenberg. Arch. Math. (Basel) 28 (1977), 253-254.

85a Faktorisierungen von Epimorphismen projektiver Ebenen. Geom. Dedicata 1 8 (1985), 281-291.

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H-ternaries and their algebraic equivaients for affine Hjelmslev planes. Vests; Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk (1973), 125-126. (Russian) Dependence of some configurational propositions in affine Hjelmslev planes. Ibid., 127. (Russian)

Affine specialization of the configuration proposition D,(8,11,14) in an AH-plane, and its algebraic equivalent. Ibid. (1975), 104-105. (Russian)

The configurational postulate DH(9,3, 17) and its algebraic equivalent. Ibid. (1978), 120-121. (Russian) The relation between certain configuration conditions in affine Hjelmslev planes. Vestii Akad. Navuk. BSSR Ser. F&.-Mat. Navuk 3 (1979), 127-128. (Russian)

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85b Distant-preserving homomorphisms between projective ring planes. Nederl. Akad. Weten- sch. P m . Ser. A 88 (= Indag. Math. 47) (1985), 443-453.

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87 Projective Barbilian spaces, I and 11. Resultate Math. 12 (1987), 222-240 and 434-449.

89 Projective geometry over finite rings. Quaderno del Sem. di Geom. Combin. no. 92. Dip. di Mat., Univ. di Roma "La Sapienza", 1989.

WEISS, H. : 87 Der Von Staudtsche Standpunkt in projektiven Veldkamp-Ebenen und Ringebenen. Diss.

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WELLER, H. : 75 Zusammenhang zwischen verbandstheoretischen Eigenschaften des Verbandes L(R3) und

arithmetischen Eigenschaften des zugehörigen Ringes. Mitt. Math. Sem. GieJ7en 118 (1975).

G. Törner Fachbereich Mathematik Universität -GH- Duisburg Postfach 101503 4100 Duisburg 1 F.R.G.

F.D. Veldkamp Mathem. Instituut R.U.U. Budapestlaan 6 postbus 80.010 3508 TA Utrecht T he Net herlands

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