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p-injectivity of simple pre-torsion modules

Published online by Cambridge University Press:  18 May 2009

K. Varadarajan
Affiliation:
The University of Calgary, Calgary, Alberta T2N 1N4, Canada University of Sydney, N.S.W. 2006, Australia
K. Wehrhahn
Affiliation:
The University of Calgary, Calgary, Alberta T2N 1N4, Canada University of Sydney, N.S.W. 2006, Australia
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V-rings and their generalisations have been studied extensively in recent years [2], [3], [5],[6], [7]. All the rings we consider will be associative rings with 1 ≠ 0 and all the modules considered will be unitary left R-modules. All the concepts will be left-sided unless otherwise mentioned. Thus by an ideal in R we mean a left ideal of R. A ring R is called a V-ring (respectively a GV-ring) if every simple (resp. simple, singular) module is injective. An R-module M is called p-injective if any homomorphism f: IM with I a principal left ideal of R can be extended to a homomorphism g: RM. A ring R is called a p-V-ring (resp. a p-V'-ring) if every simple (resp. simple, singular) module over R is p-injective. The object of the present paper is to introduce torsion theoretic generalizations of p-V-rings and prove results similar to those obtained by Yue Chi Ming about p-V-rings and p-V'-rings [6], [7]. For any M ∈ R-mod, J(M) will denote the Jacobson radical of M and Z(M) the singular submodule of M. For any λ ∈ R, we denote the left annihilator { rR| rλ =0 } of λ in R by l(λ).

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

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