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Geometrical maps in ring affine geometries

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The aim of this work is to investigate the validity of the fundamental theorem of affine geometry over rings with an invariant basis number.

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References

  1. E. Artin, Geometric Algebra, Princeton Univ. Press, Princeton, New Jersey (1957).

    MATH  Google Scholar 

  2. A. K. Bach, Teilverhältnise in affine Räumen über Modulen, Preprint, Univ. Minz. (1997).

  3. W. Benz, Vorlesungen über Geometrie der Algebren, Springer-Verlag, Berlin–New York (1973).

    Book  MATH  Google Scholar 

  4. M. Berger, Geometrie, Vols. 1, 2. CEDIC, Paris; Fernand Nathan, Paris (1977).

  5. A. Brezuleanu and D.-C. Rǎdulescu, About the collineations on open subsets of R m and p m(R), Preprint Ser. Math. No. 71/1981, INCREST, Bucharest (1981).

    Google Scholar 

  6. A. Brezuleanu and D.-C. Rǎdulescu, Characterizing lineations on open subsets of projective spaces, Preprint Ser. Math. No. 58/1982, INCREST, Bucharest (1982).

    Google Scholar 

  7. A. Brezuleanu and D.-C. Rǎdulescu, “About full injective lineations,” J. Geometry, 23, 85–60 (1984).

    Article  Google Scholar 

  8. F. Buekenhiat, Hanbook of Incidence Geometry, Elsevier, North-Holland (1995).

    Google Scholar 

  9. D. S. Carter and A. Vogt, “Collinearity-preserving functions between Desarguesian planes,” Mem. Am. Math. Soc., 27, No. 235, 1–46 (1980).

    MathSciNet  Google Scholar 

  10. P. M. Cohn, “Some remarks on the invariant basis property,” Topology, 5, 215–228 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  11. P. M. Cohn, Free Rings and Their Relations, Academic Press, London (1977).

    Google Scholar 

  12. C.-A. Faure, “Morphisms of projective spaces over rings,” Adv. Geom., 4, No. 1, 19–31 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  13. T. M. Gelashvili and A. A. Lashkhi, “Coset lattices and the fundamental theorem of affine geometry for Lie algebras,” Proc. A. Razmadze Math. Inst., 119, 43–58 (1999).

    MathSciNet  MATH  Google Scholar 

  14. M. Greferath, “Global-affine morphisms of projective lattice geometries,” Results Math., 24, Nos. 1–2, 76–83 (1993).

    MathSciNet  MATH  Google Scholar 

  15. H. Havlichek, “A generalization of Brauer’s theorem on linear mapping,” Mitt Math. Sem. Gieben, 217, 27–41 (1994).

    Google Scholar 

  16. I. Kaplansky, “Elementary divisors and modules,” Trans. Am. Math. Soc., 66, 464–491 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  17. T. G. Kvirikashvili and A. A. Lashkhi, “Perspective maps for modules,” Mat. Zametki, 77, No. 6, 946–947 (2005).

    Article  MathSciNet  Google Scholar 

  18. T. G. Kvirikashvili and A. A. Lashkhi, “Affine geometry of modules over a ring with an invariant basis number,” J. Math. Sci., 137, No. 5, 5161–5173 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  19. T. G. Kvirikashvili and A. A. Lashkhi, “Perspective maps of modules,” J. Math. Sci., 139 (2007).

  20. T. G. Kvirikashvili and A. A. Lashkhi, “Affine geometry of modules over a ring with invariant basis number,” Mat. Zametki, 82, No. 6, 838–849 (2007).

    Article  MathSciNet  Google Scholar 

  21. T. Kvirikashvili and A. Lashkhi, “On the fundamental theorem of geometric algebra over FSrings,”Commun. Algebra, 43, No. 4 (2007).

    Google Scholar 

  22. T. G. Kvirikashvili and A. A. Lashkhi, “On the fundamental theorem of geometric algebra over SF-rings,” Commun. Algebra, 36, No. 9, 3564–3573 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  23. K. J. Lambek, Rings and Modules, Waltham, Massachusetts, Toronto (1966).

    MATH  Google Scholar 

  24. A. A. Lashkhi, “The fundamental theorem of projective geometry for modules and Lie algebras,” J. Sov. Math., 42, No. 5, 1992–2008 (1988).

    Article  Google Scholar 

  25. A. A. Lashkhi, “General geometric lattices and projective geometry of modules,” J. Math. Sci., 74, No. 3, 1044–1077 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. A. Lashkhi, “Harmonic maps of modules,” Mat. Zametki, 47, No. 1, 161–163 (1990).

    MathSciNet  MATH  Google Scholar 

  27. A. A. Lashkhi, “Harmonic maps over rings,” Georgian Math. J., 4, No. 1, 41–64 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. A. Lashkhi, “Coordinatization of PI-geometrical lattices,” Dokl. Ross. Akad. Nauk 409, No. 2 (2006).

    Google Scholar 

  29. A. A. Lashkhi, “Ring geometries and their related lattices,” J. Math. Sci., 144, No. 2, 3960–3967 (2007).

    Article  MathSciNet  Google Scholar 

  30. W. G. Leavitt, “The module type of a ring,” Trans. Am. Math. Soc., 103, 113–130 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  31. W. Leisner, “Affine Barbelian-benenm, I,” J. Geometry, 6, 31–75 (1975).

    Article  Google Scholar 

  32. W. Leisner, “Affine Barbelian-benenm, II,” J. Geometry, 6, 105–129 (1975).

    Article  MathSciNet  Google Scholar 

  33. J. Lelong-Ferrand, Les Fondements de la Géométrie, Press Univ. de France (1985).

  34. J. Lipman, “Definition of affine geometry by a group of transformations,” Can. Math. Bull., 4, 265–278 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  35. C. C. Perelli, “Images of affine parallel structures by semi-collineations,” Rend. Sem. Math. Brescia, 8, 97–109 (1984).

    Google Scholar 

  36. P. Scherk, “On the fundamental theorem of affine geometry,” Can. Math. Bull., 5, 67–69 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Schmidt, Subgroup Lattices of Groups, Expos. Math., 14, de Gruyter, Berlin (1994).

  38. S. E. Schmidt, Grundlegungen zu einer allgemeinen affinen Geometrie, Birkhäuser, Basel (1995).

    Book  MATH  Google Scholar 

  39. S. E. Schmidt and S. Weller, “Fundamentalsatz für affine Räume über Moduln,” Results Math., 30, Nos. 1–2, 151–159 (1996).

    MathSciNet  MATH  Google Scholar 

  40. J. C. Shepherdson, “Inverses and zero divisors in matrix rings,” Proc. London Math. Soc. (3), 1, 71–85 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  41. Z. Zich, “Lineare Abbildungen in reelen Vektorräumen,” Math. Phys. Sem. Ber., 30, 167–176 (1983).

    Google Scholar 

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Correspondence to T. Kvirikashvili.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 74, Proceedings of the International Conference “Modern Algebra and Its Applications” (Batumi, 2010), Part 1, 2011.

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Kvirikashvili, T., Lashkhi, A. Geometrical maps in ring affine geometries. J Math Sci 186, 759–765 (2012). https://doi.org/10.1007/s10958-012-1030-x

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