GENERALIZED COFINITELY SEMIPERFECT MODULES

Abstract

In the present paper, we define generalized (amply) cofinitely supplemented modules, and generalized ⊕-cofinitely supplemented modules are defined as a generalization of (amply) cofinitely supplemented modules and ⊕-cofinitely supplemented modules, respectively, and show, among others, the following results: (1) The class of generalized cofinitely supplemented modules is closed under taking homomorphic images, generalized covers and arbitrary direct sums. (2) Any finite direct sum of generalized ⊕-cofinitely supplemented modules is a generalized ⊕-cofinitely supplemented module. (3) M is a generalized cofinitely semiperfect module if and only if M is a generalized cofinitely supplemented -module by supplements which have generalized projective covers.

Keywords

• Cofinite submodule, generalized supplement submodule, generalized projective cover, semiperfect module

References

1. R. Alizade, G. Bilhan and P.F. Smith, Modules whose maximal submodules have supplements, Comm. Algebra, 29 (2001), 2389-2405.
2. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York 1992.
3. G. Azumaya, Some characterizations of semiperfect rings and modules, in Ring Theory, edited by S.K. Jain and S. T. Rizvi, Proc. Biennial Ohio-Denison Conf., May 1992, World Scientific Publ., Singapore, 1993, 28-40.
4. E. B¨uy¨uka¸sık and C. Lomp, On a recent generalization of semiperfect rings, to appear in Bull. Aust. Math. Soc., 78(2) (2008), 317-325.
5. H. C¸ alı¸sıcı and A. Pancar, Cofinitely semiperfect Modules, Sib. Math. J., 46(2) (2005), 359-363.
6. F. Kasch, Modules and Rings, Academic Press, London 1982.
7. M. T. Ko¸san, ⊕-Cofinitely supplemented modules, Commun. Fac. Sci. Univ. Ank. Series A1, 53(1)(2004), 21-32.
8. A. Leghwel, T. Ko¸san, N. Agayev and A. Harmancı, Duo modules and duo rings, Far East J. Math., 20(3) (2006), 341-346.

9. S.H. Mohammed and B. J. M¨uller, Continous and Discrete Modules, London Math. Soc., LN 147, Cambridge Univ. Press, 1990.
10. P.F. Smith, Modules for which every submodule has a unique closure, in Ring Theory (Editors, S.K. Jain and S.T. Rizvi), World Sci. (Singapore, 1993), pp.302-313.
11. P.F. Smith, Finitely generated supplemented modules are amply supplemented, The Arab. J. Sci. and Eng., 25 (2000), 69-79.
12. Y. Wang and N. Ding, Generalized supplemented modules, Taiwanese J. Math., (6) (2006), 1589-1601.
13. X. Wue, Characterizations of semiperfect and perfect rings, Publications Matem`atiques, 40 (1996), 115-125.
14. R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991. M. Tamer Ko¸san
15. Department of Mathematics, Faculty of Science, Gebze Institute of Technology C¸ ayırova Campus 41400 Gebze-Kocaeli, Turkey email: [e-mail]