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GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES

Year 2008, Volume: 3 Issue: 3, 125 - 134, 01.06.2008

Abstract

In 2004, S.H.Mohamed and B.J.Müller defined generalized projectivity (dual ojectivity) as follows: given modules A and B, A is generalized B-projective (B-dual ojective) if, for any homomorphism f : A → X and any epimorphism g : B → X, there exist decompositions A = A0 ⊕ A00, B = B0⊕B00, a homomorphism ϕ : A0 → B0 and an epimorphism ψ : B00 → A00 such that g ◦ ϕ = f|A0 and f ◦ ψ = g|B0 . Generalized projectivity plays an important role in the study of direct sums of lifting modules. Since the structure of generalized projectivity is complicated, it is difficult to determine whether generalized projectivity is inherited by (finite) direct sums. This problem is not easy even in the case that each module is quasi-discrete. In this paper we consider this problem.

References

  • F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer Verlag, 1973.
  • Y. Baba and M. Harada, On almost M-projectives and almost M-injectives, Tsukuba J. Math., 14 (1990), 53-69.
  • J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules. Supplements and Projectivity in Module Theory, Frontiers in Math., Boston, Birkh¨auser, L. Ganesan and N. Vanaja, Modules for which every submodule has a unique coclosure, Comm. Algebra, 30 (2002), 2355–2377.
  • K. Hanada, Y. Kuratomi and K. Oshiro, On direct sums of extending modules and internal exchange property, J. Algebra, 250 (2002), 115-133.
  • M. Harada, Factor categories with applications to direct decomposition of mod- ules, LN in Pure and Applied Math. 88, Marcel Dekker, Inc., New York, 1983.
  • D. Keskin, On lifting modules, Comm. Algebra, 28 (2000), 3427-3440.
  • Y. Kuratomi, On direct sums of lifting modules and internal exchange property, Comm. Algebra., 33 (2005), 1795-1804.
  • Y. Kuratomi and C. Chaehoon, Lifting modules over right perfect rings, Comm. Algebra., 35 (2007), 3103-3109.
  • S.H. Mohamed and B.J. M¨uller, Continuous and Discrete Modules, London Math. Soc., LN 147, Cambridge Univ. Press, Cambridge, 1990.
  • S.H. Mohamed and B.J. M¨uller, Cojective modules, J. Egyptian Math. Soc. 12 (2004), 83-96.
  • K. Oshiro, Semiperfect modules and quasi-semiperfect modules, Osaka J. Math. (1983), 337-372.
  • K. Oshiro, Theories of Harada in Artinian Rings and applications to classical Artinian Rings, International Symposium on Ring Theory, Trends in Math., Birkh¨auser, (2001), 279-301.
  • R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publications, 1991. Yosuke Kuratomi
  • Kitakyushu National College of Technology, Sii, Kokuraminami-ku, Kitakyushu, Japan E-mail: kuratomi@kct.ac.jp
Year 2008, Volume: 3 Issue: 3, 125 - 134, 01.06.2008

Abstract

References

  • F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer Verlag, 1973.
  • Y. Baba and M. Harada, On almost M-projectives and almost M-injectives, Tsukuba J. Math., 14 (1990), 53-69.
  • J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules. Supplements and Projectivity in Module Theory, Frontiers in Math., Boston, Birkh¨auser, L. Ganesan and N. Vanaja, Modules for which every submodule has a unique coclosure, Comm. Algebra, 30 (2002), 2355–2377.
  • K. Hanada, Y. Kuratomi and K. Oshiro, On direct sums of extending modules and internal exchange property, J. Algebra, 250 (2002), 115-133.
  • M. Harada, Factor categories with applications to direct decomposition of mod- ules, LN in Pure and Applied Math. 88, Marcel Dekker, Inc., New York, 1983.
  • D. Keskin, On lifting modules, Comm. Algebra, 28 (2000), 3427-3440.
  • Y. Kuratomi, On direct sums of lifting modules and internal exchange property, Comm. Algebra., 33 (2005), 1795-1804.
  • Y. Kuratomi and C. Chaehoon, Lifting modules over right perfect rings, Comm. Algebra., 35 (2007), 3103-3109.
  • S.H. Mohamed and B.J. M¨uller, Continuous and Discrete Modules, London Math. Soc., LN 147, Cambridge Univ. Press, Cambridge, 1990.
  • S.H. Mohamed and B.J. M¨uller, Cojective modules, J. Egyptian Math. Soc. 12 (2004), 83-96.
  • K. Oshiro, Semiperfect modules and quasi-semiperfect modules, Osaka J. Math. (1983), 337-372.
  • K. Oshiro, Theories of Harada in Artinian Rings and applications to classical Artinian Rings, International Symposium on Ring Theory, Trends in Math., Birkh¨auser, (2001), 279-301.
  • R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publications, 1991. Yosuke Kuratomi
  • Kitakyushu National College of Technology, Sii, Kokuraminami-ku, Kitakyushu, Japan E-mail: kuratomi@kct.ac.jp
There are 14 citations in total.

Details

Other ID JA98GY57CP
Journal Section Articles
Authors

Yosuke Kuratomi This is me

Publication Date June 1, 2008
Published in Issue Year 2008 Volume: 3 Issue: 3

Cite

APA Kuratomi, Y. (2008). GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES. International Electronic Journal of Algebra, 3(3), 125-134.
AMA Kuratomi Y. GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES. IEJA. June 2008;3(3):125-134.
Chicago Kuratomi, Yosuke. “GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES”. International Electronic Journal of Algebra 3, no. 3 (June 2008): 125-34.
EndNote Kuratomi Y (June 1, 2008) GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES. International Electronic Journal of Algebra 3 3 125–134.
IEEE Y. Kuratomi, “GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES”, IEJA, vol. 3, no. 3, pp. 125–134, 2008.
ISNAD Kuratomi, Yosuke. “GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES”. International Electronic Journal of Algebra 3/3 (June 2008), 125-134.
JAMA Kuratomi Y. GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES. IEJA. 2008;3:125–134.
MLA Kuratomi, Yosuke. “GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES”. International Electronic Journal of Algebra, vol. 3, no. 3, 2008, pp. 125-34.
Vancouver Kuratomi Y. GENERALIZED PROJECTIVITY OF QUASI-DISCRETE MODULES. IEJA. 2008;3(3):125-34.