Finitely Generated Modules over Semilocal Rings and Characterizationsof (Semi-)Perfect Rings
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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.
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.
참고문헌 (12)
- A. Amini, "On generalized perfect rings", Comm. Algebra, 35(3), pp.953~963 (2007)
- F. W. Anderson, "Rings and Categories of Modules, Springer-Verlag" (1992), Berlin-Heidelberg-New York
- Y. Baba, "Artinian Rings and Related Topics", Lecture
- H. Bass, "Finitistic Dimension and Homological Generalization of Semiprimary Rings", Trans. Amer. Math, 95, pp.466~486 (1960)
- J. Clark, "Lifting modules" (2007), Birkhauser Boston, Boston
- L. Ganesan, "Strongly discrete modules", Comm. Algebra, 35(3), pp.897~913 (2007)
- K. Hanada, "On direct sums of extending modules and internal exchange property", J. Algebra, 250, pp.115~133 (2002)
- Y. Kuratomi, "Lifting modules over right perfect rings", Comm. Algebra, 35(10), pp.3103~3109 (2007)
- C. Lomp, "On semilocal modules and rings", Comm. Algebra, 27(4), pp.1921~1935 (1999)
- E. Mares, "Semiperfect modules", Math. Z, 82, pp.347~360 (1963)
- I. I. Sakhajev, "On the weak dimension of modules, rings, algebras", Projectivity of flat modules Izv. Vyssh. Uchebn. Zaved. Mat, 2, pp.152~157 (1965)
- R. Wisbauer, "Foundations of Modules and Ring Theory" (1991), Gordon and Breach Science Publishers