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Finitely Generated Modules over Semilocal Rings and Characterizations of (Semi-)Perfect Rings
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  • Journal title : Kyungpook mathematical journal
  • Volume 48, Issue 1,  2008, pp.143-154
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2008.48.1.143
 Title & Authors
Finitely Generated Modules over Semilocal Rings and Characterizations of (Semi-)Perfect Rings
Chang, Chae-Hoon;
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 Abstract
Lomp [9] has studied finitely generated projective modules over semilocal rings. He obtained the following: finitely generated projective modules over semilocal rings are semilocal. We shall give necessary and sufficient conditions for finitely generated modules to be semilocal modules. By using a lifting property, we also give characterizations of right perfect (semiperfect) rings. Our main results can be summarized as follows: (1) Let M be a finitely generated module. Then M has finite hollow dimension if and only if M is weakly supplemented if and only if M is semilocal. (2) A ring R is right perfect if and only if every flat right R-module is lifting and every right R-module has a flat cover if and only if every quasi-projective right R-module is lifting. (3) A ring R is semiperfect if and only if every finitely generated flat right R-module is lifting if and only if RR satisfies the lifting property for simple factor modules.
 Keywords
perfect ring;semiperfect ring;semilocal ring;finitely generated module;lifting module;amply supplemented module;
 Language
English
 Cited by
 References
1.
A. Amini, B. Amini, M. Ershad, and H. Sharif, On generalized perfect rings, Comm. Algebra ,35(3)(2007), 953-963. crossref(new window)

2.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, Berlin-Heidelberg-New York, (1992).

3.
Y. Baba and K. Oshiro, Artinian Rings and Related Topics, Lecture Note.

4.
H. Bass, Finitistic Dimension and Homological Generalization of Semiprimary Rings, Trans. Amer. Math., 95(1960), 466-486. crossref(new window)

5.
J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Birkhauser Boston, Boston (2007).

6.
L. Ganesan and N. Vanaja, Strongly discrete modules, Comm. Algebra, 35(3)(2007), 897-913. crossref(new window)

7.
K. Hanada, Y. Kuratomi, and K. Oshiro, On direct sums of extending modules and internal exchange property, J. Algebra, 250(2002), 115-133. crossref(new window)

8.
Y. Kuratomi and C. Chang, Lifting modules over right perfect rings, Comm. Algebra, 35(10)(2007), 3103-3109. crossref(new window)

9.
C. Lomp, On semilocal modules and rings, Comm. Algebra, 27(4)(1999), 1921-1935. crossref(new window)

10.
E. Mares, Semiperfect modules, Math. Z., 82(1963), 347-360. crossref(new window)

11.
I. I. Sakhajev, On the weak dimension of modules, rings, algebras. Projectivity of flat modules, Izv. Vyssh. Uchebn. Zaved. Mat., 2(1965), 152-157.

12.
R. Wisbauer, Foundations of Modules and Ring Theory, Gordon and Breach Science Publishers (1991).