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On a class of QI-rings

Published online by Cambridge University Press:  18 May 2009

S. K. Jain ,
S. R. López-Permouth  and
Surjeet Singh
S. K. Jain
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701
S. R. López-Permouth
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701
Surjeet Singh
Affiliation:
Department of Mathematics, Kuwait University, Kuwait
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The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right R-module if and only if R is right artinian and every indecomposable projective right R-module is uniform and weakly R-injective. We show that in the above characterization the requirement that indecomposable projective right R-modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right QI-rings. A ring R is said to be a right QI-ring if every quasi-injective right R-module is injective. QI-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 1 , January 1992 , pp. 75 - 81
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

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