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Finitely Generated Modules over Semilocal Rings and Characterizations of (Semi-)Perfect Rings

Chang, Chae-Hoon

  • Received : 2007.11.13
  • Published : 2008.03.31

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

References

  1. A. Amini, B. Amini, M. Ershad, and H. Sharif, On generalized perfect rings, Comm. Algebra ,35(3)(2007), 953-963. https://doi.org/10.1080/00927870601115880
  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. https://doi.org/10.1090/S0002-9947-1960-0157984-8
  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. https://doi.org/10.1080/00927870601115815
  7. K. Hanada, Y. Kuratomi, and K. Oshiro, On direct sums of extending modules and internal exchange property, J. Algebra, 250(2002), 115-133. https://doi.org/10.1006/jabr.2001.9089
  8. Y. Kuratomi and C. Chang, Lifting modules over right perfect rings, Comm. Algebra, 35(10)(2007), 3103-3109. https://doi.org/10.1080/00927870701405132
  9. C. Lomp, On semilocal modules and rings, Comm. Algebra, 27(4)(1999), 1921-1935. https://doi.org/10.1080/00927879908826539
  10. E. Mares, Semiperfect modules, Math. Z., 82(1963), 347-360. https://doi.org/10.1007/BF01111401
  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).