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Weakly-injective modules over hereditary noetherian prime rings

Published online by Cambridge University Press:  09 April 2009

S. K. Jain  and
S. R. López-Permouth
S. K. Jain
Affiliation:
Ohio University, Athens, Ohio 45701
S. R. López-Permouth
Affiliation:
Ohio University, Athens, Ohio 45701
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Abstract

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A module M is said to be wealdy-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N ⊂ X. In this paper we investigate weakly-injective modules over bounded hereditary noetherian prime rings. In particular we show that torsion-free modules over bounded hnp rings are always wealdy-injective, while torsion modules with finite Goldie dimension are weakly-injective only if they are injective.

As an application, we show that weakly-injective modules over bounded Dedekind prime rings have a decomposition as a direct sum of an injective module B, and a module C satisfying that if a simple module S is embeddable in C then the (external) direct sum of all proper submodules of the injective hull of S is also embeddable in C. Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective module has such a decomposition.

Keywords

bounded hereditary noetherian prime ringsweakly-injective modules.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 3 , June 1995 , pp. 287 - 297
Copyright
Copyright © Australian Mathematical Society 1995

References

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