Extending modules with injective or semisimple summands
type: Item
instance of: scholarly article
title: Extending modules with injective or semisimple summands
zbMATH Open document ID: 0934.16003
DOI: 10.1016/S0022-4049(96)00156-9
author: Catarina Santa-Clara
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-27T00:00:00Z
review text: A module \(M\) is called an extending module if every submodule of \(M\) is essential in a direct summand. A natural question that arises in the study of extending modules is to determine necessary and sufficient conditions for a direct sum of extending modules to be extending. The paper under review addresses this question, and gives satisfactory answers in some special cases. For example, necessary and sufficient conditions are obtained for a direct sum of an extending module and an injective module to be extending, or for a direct sum of a module with the finite exchange property and a semisimple module to be extending. These conditions involve some relative injectivity property between the summands. Finally, it is proved that a ring \(R\) satisfies the property that every direct sum of an extending right \(R\)-module and a semisimple right \(R\)-module is extending if and only if \(R/\text{Soc}(R_R)\) is a right Noetherian right \(V\)-ring.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 1308163
zbMATH Keywords: extending modules
zbMATH Keywords: injective modules
zbMATH Keywords: semisimple modules
zbMATH Keywords: finite exchange property
zbMATH Keywords: direct summands
zbMATH Keywords: direct sums
zbMATH Keywords: relative injectivity
zbMATH Keywords: right Noetherian right \(V\)-rings
Wikidata QID: Q114122620
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308163