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Modules which are invariant under monomorphisms of their injective hulls

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  09 April 2009

A. Alahmadi ,
N. Er  and
S. K. Jain
A. Alahmadi
Affiliation:
Department of MathematicsOhio UniversityAthens, OH 45701USA e-mail: [email protected]
N. Er
Affiliation:
Department of MathematicsThe Ohio State University-NewarkOH 43055, USA
S. K. Jain
Affiliation:
Department of MathematicsThe Ohio State University-NewarkOH 43055, USA
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Abstract

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In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of R into R(N)R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is quasi-injective. The following results are obtained: M is quasi-injective if and only if M is pseudo-injective and M2 is CS. Furthermore, if M is a direct sum of uniform modules, then M is quasi-injective if and only if M is pseudo-injective. As a consequence of this it is shown that over a right Noetherian ring R, quasi-injective modules are precisely pseudo-injective CS modules.

MSC classification

Secondary: 16D50: Injective modules, self-injective rings 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 3 , December 2005 , pp. 349 - 360
Copyright
Copyright © Australian Mathematical Society 2005

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