Rings whose cyclics have finite Goldie dimension
type: Item
instance of: scholarly article
title: Rings whose cyclics have finite Goldie dimension
zbMATH Open document ID: 0766.16007
DOI: 10.1016/0021-8693(92)90147-E
author: A. H. Al-Huzali
author: Sergio R. López-Permouth
author: S. K. Jain
published in: Journal of Algebra
publication date: 1993-03-28T00:00:00Z
review text: Given two right modules \(M\) and \(N\), the authors say that \(M\) is weakly \(N\)-injective if for every homomorphism \(\varphi:N\to E(M)\) there exists a submodule \(X\subseteq E(M)\) which is isomorphic to \(M\) satisfying \(\varphi(N)\subseteq X\). And \(M\) is weakly injective if it is weakly \(N\)- injective for every finitely generated right module \(N\). The authors prove that a ring \(R\) satisfies the property that every cyclic right \(R\)- module has finite Goldie dimension if and only if every direct sum of (weakly) injective right \(R\)-modules is weakly injective; if and only if every direct sum of indecomposable injective (weakly injective) right \(R\)-modules is weakly \(R\)-injective.
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 130477
zbMATH Keywords: weakly \(N\)-injective
zbMATH Keywords: cyclic right \(R\)-module
zbMATH Keywords: injective right \(R\)- modules
zbMATH Keywords: finitely generated right module
zbMATH Keywords: finite Goldie dimension
zbMATH Keywords: direct sum
reviewed by: Xue Weimin
MaRDI profile type: Publication
cites work: Modules with decompositions that complement direct summands
cites work: Modules whose quotients have finite Goldie dimension
cites work: Rings whose cyclics are essentially embeddable in projective modules
cites work: On a class of QI-rings
cites work: Rings whose cyclic modules have finitely generated socle
cites work: Dual generalizations of the Artinian and Noetherian conditions
full work available at URL: 0021-8693(92)90147-e
OpenAlex ID: W2042323164
description: scientific article; zbMATH DE number 130477