Abstract
In Keppens (Innov. Incidence Geom. 15: 119–139, 2017) we gave a state of the art concerning “projective planes” over finite rings. The current paper gives a complementary overview for “affine planes” over rings (including the important subclass of desarguesian affine Klingenberg and Hjelmslev planes). No essentially new material is presented here but we give a summary of known results with special attention to the finite case, filling a gap in the literature.
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References
Arnold, H.J.: Die Geometrie der Ringe im Rahmen allgemeiner affiner Strukturen, Hamburger Mathematische Einzelschriften. Neue Folge. Heft 4 (1971)
Arnold, H.J.: A way to the geometry of rings. J. Geom. 1, 155–167 (1971)
Bach, A.K.: Teilverhältnisse in affinen Räumen über Moduln. Beiträge Algebra Geom. 38, 385–398 (1997)
Bacon, P.Y.: Desarguesian Klingenberg planes. Trans. Am. Math. Soc. 241, 343–355 (1978)
Barbilian, D.: Zur Axiomatik der projektiven ebenen Ringgeometrien I and II. Jahresbericht Deutsch. Math. Verein 50, 179–229 (1940)
Barbilian, D.: Zur Axiomatik der projektiven ebenen Ringgeometrien I and II. Jahresbericht Deutsch. Math. Verein 51, 34–76 (1941)
Bartnicka, E., Matraś, A.: Free Cyclic Submodules in the Context of the Projective Line. Results Math. 70, 567–580 (2016)
Batbedat, A.: Espaces de Permutti généralises. Rend. Ist. Mat. Univ. Trieste 8, 14–21 (1976)
Batbedat, A.: Le problème de Pizzarello sur les sous-espaces géométriques d’un espace de Permutti. Rend. Ist. Mat. Univ. Trieste 8, 22–34 (1976)
Benz, W.: \(\Omega \)-Geometrie und Geometrie von Hjelmslev. Math. Ann. 164, 118–123 (1966)
Benz, W.: Ebene Geometrie über einem Ring. Math. Nachr. 59, 163–193 (1974)
Benz, W.: On Barbilian domains over commutative rings. J. Geom. 12, 146–151 (1979)
Bini, G., Flamini, F.: Finite Commutative Rings and Their Applications. Kluwer, Boston (2002)
Bisztriczky, T., Lorimer, J.W.: Axiom systems for affine Klingenberg spaces. In: Combinatorics ’88, (Ravello, 1988), Vol. 1, pp. 185–200. Mediterranean, Rende (1991)
Bisztriczky, T., Lorimer, J.W.: On hyperplanes and free subspaces of affine Klingenberg spaces. Aequationes Math. 48, 121–136 (1994)
Bisztriczky, T., Lorimer, J.W.: Subspace operations in affine Klingenberg spaces. Bull. Belg. Math. Soc. Simon Stevin 2, 99–108 (1995)
Blunck, A., Havlicek, H.: The connected components of the projective line over a ring. Adv. Geom. 1, 107–117 (2001)
Buekenhout, F., Cameron, P.: Projective and affine geometry over division rings. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, pp. 27–62. Elsevier, Amsterdam (1995)
Chen, H.: Rings Related to Stable Range Conditions, volume 11 of Series in Algebra. World Scientific, Singapore (2011)
Cohn, P.M.: On the structure of the \(GL_{2}\) of a ring. Publ. Math. I.H.E.S 30, 365–413 (1966)
Cohn, P.M.: Free Rings and Their Relations. Academic Press, London (1971)
De Clerck, F., Van Maldeghem, H.: Some classes of rank 2 geometries. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, pp. 433–475. Elsevier, Amsterdam (1995)
Dorn, G.: Affine Geometrie über Matrizenringen. Mitt. Math. Semin. Giessen 109, 120p (1974)
Emel’chenkov, E.P.: \((\Pi , l)\) collineations of AH-planes, Modern geometry, Leningrad. Gos. Ped. Inst. (in russian). 58–60 (1978)
Everett, C.J.: Affine geometries of vector spaces over rings. Duke Math. J. 9, 873–878 (1942)
Eugeni, F., Galiè, E.: Sui piani costruiti su anelli, pp. 143–162. Università di Teramo, Italy, Dipartimento M.E.T. (1991)
Faulkner, J.R.: Barbilian planes. Geom. Ded. 30, 125–181 (1989)
Faulkner, J.R.: The role of nonassociative algebra in projective geometry. Graduate Studies in Mathematics, vol. 159. AMS, Providence (2014)
Groze, V., Vasiu, A.: Affine structures over an arbitrary ring. Studia Univ. Babes-Bolyai Math. 25, 28–31 (1980)
Honold, T., Landjev, I.: Codes over rings and ring geometries, In: Storme, L., de Beule J. (eds.) Current Research Topics in Galois geometry, vol. 7, pp. 159–184. Nova Science Publishers, New–York (2011)
Iordănescu, R.: The geometrical Barbilian’s work from a modern point of view. Balkan J. Geom. Appl. 1, 31–36 (1996)
Keppens, D.: 50 years of Finite Geometry, the “Geometry over finite rings” part. Innov. Incidence Geom. 15, 119–139 (2017)
Klingenberg, W.: Projektive und affine Ebenen mit Nachbarelementen. Math. Z. 60, 384–406 (1954)
Klingenberg, W.: Desarguessche Ebenen mit Nachbarelementen. Abh. Math. Sem. Univ. Hamburg 20, 97–111 (1955)
Klingenberg, W.: Projektive Geometrien mit Homomorphismus. Math. Ann 132, 180–200 (1956)
Kreuzer, A.: Hjelmslevräume. Result. Math. 12, 148–156 (1987)
Kreuzer, A.: Free modules over Hjelmslev rings in which not every maximal linearly independent subset is a basis. J. Geom. 45, 105–113 (1992)
Kvirikashvili, T., Lashkhi, A.: Geometrical maps in ring affine geometries. J. Math. Sci. (N.Y.) 186, 759–765 (2012)
Lam, T.Y.: A First Course in Non-commutative Rings, Graduate Texts in Mathematics, vol. 131. Springer, New York (2001)
Lashkhi, A., Kvirikashvili, T.: Affine geometry of modules over a ring with an invariant basis number. J. Math. Sci. (N.Y.) 137, 5161–5173 (2006)
Lantz, D.: Uniqueness of Barbilian domains. J. Geom. 15, 21–27 (1981)
Leissner, W.: Affine Barbilian-Ebenen I. J. Geom. 6, 31–57 (1975)
Leissner, W.: Affine Barbilian-Ebenen II. J. Geom. 6, 105–129 (1975)
Leissner, W.: Barbilianbereiche. In: Arnold, H.J., Benz, W., Wefelscheid, H. (eds.) Beiträge zur Geometrischen Algebra, pp. 219–224. Birkhäuser, Basel Stuttgart (1977)
Leissner, W.: On classifying affine Barbilian spaces. Result. Math. 12, 157–171 (1987)
Leissner, W.: Rings of stable rank 2 are Barbilian rings. Result. Math. 20, 530–537 (1991)
Leissner, W., Severin, R., Wolf, K.: Affine geometry over free unitary modules. J. Geom. 25, 101–120 (1985)
Lorimer, J.W.: Coordinate theorems for affine Hjelmslev planes. Ann. Mat. Pura ed Applicata 105, 170–191 (1975)
Lorimer, J.W.: The fundamental theorem of Desarguesian affine Hjelmslev planes. Mitt. Math. Sem. Giessen 119, 6–14 (1975)
Lorimer, J.W.: Affine Hjelmslev rings and planes. Annals of Discr. Math. 37, 265–276 (1988)
Lorimer, J.W.: What is a collineation of the integer plane? Am. Math. Mon. 103, 687–691 (1996)
Lorimer, J.W., Lane, N.D.: Desarguesian affine Hjelmslev planes. J. für die reine und angew. Math. 1, 336–352 (1975)
Lück, H.H.: Projektive Hjelmslevräume. J. Reine Angew. Math. 243, 121–158 (1970)
Lüneburg, H.: Affine Hjelmslev-Ebenen mit transitiver Translationsgruppe. Math. Z. 79, 260–288 (1962)
Magnus, T.D.: Faulkner geometry. Geom. Ded. 59, 1–28 (1996)
Mäurer, H., Nolte, W.: A characterization of pappian affine Hjelmslev planes. Ann. Discrete Math. 37, 281–291 (1988)
Nechaev, A.A.: Finite rings with applications, In: Handbook of Algebra, vol. 5, Elsevier, New York (2008)
Nolte, W.: Pappusche affine Klingenbergebenen. J. Geom. 52, 152–158 (1995)
Permutti, R.: Geometria affine su di un anello, Atti Accad. Naz. Lincei, Mem. Cl. Sc. Fis. Mat. Nat. 8, 259–287 (1967)
Pizzarello, G.: Sugli spazi affini sopra un anello. Rend. Ist. Mat. Univ. Trieste 1, 98–111 (1969)
Planat, M., Saniga, M., Kibler, M.R.: Quantum entanglement and projective ring geometry. SIGMA 2, 1–14 (2006)
Radó, F.: Affine Barbilian structures. J. Geom. 14, 75–102 (1980)
Salunke, J.N.: On commutativity of finite rings. Bulletin of the Marathwada Math. Soc. 13, 39–47 (2012)
Schmidt, S.E., Steinitz, R.: The coordinatization of affine planes by rings. Geom. Ded. 62, 299–317 (1996)
Schmidt, S.E., Weller, S.: Fundamentalsatz für affine Räume über Moduln. Results Math. 30, 151–159 (1996)
Seier, W.: Der kleine Satz von Desargues in affinen Hjelmslev-Ebenen. Geom. Ded. 3, 215–219 (1974)
Törner, G., Veldkamp, F.D.: Literature on geometry over rings. J. Geom. 42, 180–200 (1991)
Veldkamp, F.D.: Geometry over Rings. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, pp. 1033–1084. Elsevier, Amsterdam (1995)
Veldkamp, F.D.: \(n\)-Barbilian domains. Results in Math. 23, 177–200 (1993)
Wan, Z.-X.: Lectures on Finite Fields and Galois rings. World Scientific, River Edge, NJ (2003)
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To the memory of Prof. Walter Benz († 13-01-2017).
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Keppens, D. Affine planes over finite rings, a summary. Aequat. Math. 91, 979–993 (2017). https://doi.org/10.1007/s00010-017-0497-4
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DOI: https://doi.org/10.1007/s00010-017-0497-4