Some Results on δ-Semiperfect Rings and δ-Supplemented Modules

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 2,  2015, pp.289-300
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.2.289
Title & Authors
Some Results on δ-Semiperfect Rings and δ-Supplemented Modules
ABDIOGLU, CIHAT; SAHINKAYA, SERAP;

Abstract
In [9], the author extends the definition of lifting and supplemented modules to $\small{{\delta}}$-lifting and $\small{{\delta}}$-supplemented by replacing "small submodule" with "$\small{{\delta}}$-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of $\small{{\delta}}$-lifting and $\small{{\delta}}$-supplemented modules. Especially, we show that any finite direct sum of $\small{{\delta}}$-hollow modules is $\small{{\delta}}$-supplemented. On the other hand, the notion of amply $\small{{\delta}}$-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply $\small{{\delta}}$-supplemented and satisfies Descending Chain Condition (DCC) on $\small{{\delta}}$-supplemented modules and on $\small{{\delta}}$-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a $\small{{\delta}}$-semiperfect ring which satisfies DCC on $\small{{\delta}}$-small right ideals of R.
Keywords
$\small{{\delta}}$-small submodules;$\small{{\delta}}$-supplemented modules;$\small{{\delta}}$-lifting modules;amply $\small{{\delta}}$-supplemented modules;
Language
English
Cited by
References
1.
K. Al-Takhman, Cofinitely ${\delta}$-supplemented abd cofinitely ${\delta}$-semiperfect modules, International Journal of Algebra, 1(12)(2007), 601-613.

2.
F. W. Anderson and K. R. Fuller , Rings and Categories of Modules, Springer-Verlag, New York, (1974).

3.
E. Buyukasik and C. Lomp, When ${\delta}$-semiperfect rings are semiperfect, Turkish J. Math., 34(2010), 317-324.

4.
J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, (2006) Birkhauser, Basel.

5.
A. Idelhadj and R. Tribak, A dual notion of CS-modules generalization, Algebra and Number Theory (Fcz) (M. Boulagouaz and J.P. Tignol, eds.), Lecture Note of Pure and Appl. Math., 208, Marcel Dekker, New York 2000 pp. 149-155.

6.
A. Idelhadj and R. Tribak, Modules for which every submodule has a supplement that is a direct summand, Arab. J. Sci. Eng. Sect. C Theme Issues, 25(2)(2000), 179-189.

7.
A. Idelhadj and R. Tribak, On some properties of ${\oplus}$-supplemented modules, IJMMS 69(2003), 4373-4387.

8.
D. Keskin, On lifting modules, Comm. Alg., 28(7)(2000), 3427-3440.

9.
M. T. Kosan, ${\delta}$-lifting and ${\delta}$-supplemented modules, Alg. Colloq., 14(1)(2007), 53-60.

10.
M. T. Kosan and A. C. Ozcan, ${\delta}$-M-small and ${\delta}$-Harada modules, Comm. Alg., 36(2)(2008), 423-433.

11.
S. H. Mohamed and B. J. Muller, Continuous and discret modules, London Math. Soc. LNS 147 Cambridge Univ. Press, Cambridge, 1990.

12.
R. Tribak, On cofinitely lifting and cofinitely weak lifting modules, Comm. Alg., 36(12)(2008), 4448-4460.

13.
Y. Zhou, Generalizations of perfect, semiperfect and semiregular rings, Alg. Colloq., 7(3)(2000), 305-318.