Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Left perfect rings that are right perfect and a characterization of Steinitz rings

Author: William H. Rant
Journal: Proc. Amer. Math. Soc. 32 (1972), 81-84
MSC: Primary 16.50
MathSciNet review: 0289571
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A proof is given to show all flat left modules of a ring are free if and only if the ring is a local ring with a left T-nilpotent maximal ideal. We characterize left perfect rings whose radical R has the property that $ I{R^n} = \{ 0\} $ for some positive integer n if I is a finitely generated right ideal contained in R. We cite an example of a left perfect ring which does not have this property. It is shown that if the set of irreducible elements of a left perfect ring is right T-nilpotent then the ring is right perfect.

References [Enhancements On Off] (What's this?)

  • [1] H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. MR 28 #1212. MR 0157984 (28:1212)
  • [2] J.-E. Björk, Rings satisfying a minimum condition on principal ideals, J. Reine Angew. Math. 236 (1969), 112-119. MR 40 #1419. MR 0248165 (40:1419)
  • [3] B.-S. Chwe and J. Neggers, On the extension of linearly independent subsets of free modules to bases, Proc. Amer. Math. Soc. 24 (1970), 466-470. MR 40 #5652. MR 0252432 (40:5652)
  • [4] V. E. Govorov, Rings over which flat modules are free, Dokl. Akad. Nauk SSSR 144 (1962), 965-967 = Soviet Math. Dokl. 3 (1962), 836-838. MR 25 #3076. MR 0139645 (25:3076)
  • [5] B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373-387. MR 34 #4305. MR 0204463 (34:4305)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16.50

Retrieve articles in all journals with MSC: 16.50

Additional Information

Keywords: Perfect ring, Steinitz ring, T-nilpotent, local ring, radical, irreducible element
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society