Ukrainian Mathematical Journal

, Volume 63, Issue 5, pp 768–775 | Cite as

On strongly ⊕-supplemented modules

  • C. Nebiyev
  • A. Pancar

Strongly ⊕-supplemented and strongly cofinitely ⊕-supplemented modules are defined and some properties of strongly ⊕-supplemented and strongly cofinitely ⊕-supplemented modules are investigated. Let R be a ring. Then every R-module is strongly ⊕-supplemented if and only if R is perfect. The finite direct sum of ⊕-supplemented modules is ⊕-supplemented. However, this is not true for strongly ⊕-supplemented modules. Any direct sum of cofinitely ⊕-supplemented modules is cofinitely ⊕-supplemented but this is not true for strongly cofinitely ⊕-supplemented modules. We also prove that a supplemented module is strongly ⊕-supplemented if and only if every supplement submodule lies above a direct summand.


Direct Summand Left Ideal Projective Module Projective Cover Discrete Valuation Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Alizade and A. Pancar, Homoloji Cebire Giris, Ondokuz Mayıs Üniv. Fen-Edebiyat Fakültesi, Samsun (1999).Google Scholar
  2. 2.
    R. Alizade, G. Bilhan, and P. F. Smith, “Modules whose maximal submodules have supplements,” Commun. Algebra, 29, No. 6, 2389–2405 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    R. Alizade and E. Büyükaşik, “Cofinitely weak supplemented modules,” Commun. Algebra, 13, 5377–5390 (2003).CrossRefGoogle Scholar
  4. 4.
    F. V. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer (1992).Google Scholar
  5. 5.
    F. Callialp, Cebir, Sakarya Üniv. Yayin, Sakarya, No. 6 (1995).Google Scholar
  6. 6.
    T. W. Hungerford, Algebra, Springer, New York (1973).Google Scholar
  7. 7.
    F. Kasch, Modules and Rings, Academic Press, New York (1982).zbMATHGoogle Scholar
  8. 8.
    I. Kaplansky, Infinite Abelian Groups, Ann Arbor (1969).Google Scholar
  9. 9.
    I. Kaplansky, “Projective modules,” Ann. Math., 68, 372–377 (1958).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    D. Keskin, A. Harmanci, and P. F. Smith, “On ⊕-supplemented modules,” Acta Math. Hung., 83, No. 1–2, 161–169 (1999).MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. Keskin, P. F. Smith, and W. Xue, “Rings whose modules are ⊕-supplemented,” J. Algebra, 218, 470–487 (1999).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C. Lomp, “On semilocal modules and rings,” Commun. Algebra, 27, No. 4, 1921–1935 (1999).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    C. Lomp, On Dual Goldie Dimension, Ph.D. Thesis, Düsseldorf (1996).Google Scholar
  14. 14.
    S. H. Mohamed and B. J. Müller, “Continuous and discrete modules,” London Math. Soc., Cambridge Univ. Press, Cambridge, 147 (1990).Google Scholar
  15. 15.
    C. Nebiyev “Amply weak supplemented modules,” Int. J. Comput. Cognition, 3, No. 1, 88–90 (2005).Google Scholar
  16. 16.
    B. Sarath and K. Varadarajan, “Injectivity of direct sums,” Commun. Algebra, 1, 517–530 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    W. D. Sharpe and P. Vamos, Injective Modules, Cambridge Univ. Press (1972).Google Scholar
  18. 18.
    P. F. Smith, “Finitely generated supplemented modules are amply supplemented,” Arab. J. Sci. Eng., 25, No. 2, 69–79 (2000).MathSciNetGoogle Scholar
  19. 19.
    R. Wisbauer, Foundations of Module and Ring Theory, Gordon & Breach, Philadelphia (1991).zbMATHGoogle Scholar
  20. 20.
    H. Zöschinger, “Komplementierte Moduln über Dedekindringen,” J. Algebra, 29, 42–56 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    H. Zöschinger, “Komplemente als direkte Summanden,” Arch. Math., 25, 241–243 (1974).zbMATHCrossRefGoogle Scholar
  22. 22.
    H. Zöschinger, “Projektive Moduln mit endlich erzeugten Radikalfaktormoduln,” Math. Ann., 255, 199–206 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    H. Zöschinger, “Komplemente als direkte Summanden II,” Arch. Math., 38, 324–334 (1982).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • C. Nebiyev
    • 1
  • A. Pancar
    • 1
  1. 1.SamsunTurkey

Personalised recommendations