Every \(\aleph_1\)-\(\Sigma\)-CS module is \(\Sigma\)-CS.
type: Item
instance of: scholarly article
title: Every \(\aleph_1\)-\(\Sigma\)-CS module is \(\Sigma\)-CS.
zbMATH Open document ID: 1072.16005
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(R\) be an associative ring. A right \(R\)-module \(M\) is called CS when every submodule of \(M\) is essential in a direct summand. The modules \(M\) such that every direct sum of copies of \(M\) is CS are called \(\Sigma\)-CS modules and, more generally, \(M\) is \(\aleph\)-\(\Sigma\)-CS, where \(\aleph\) is an infinite cardinal, whenever every direct sum of \(\aleph\) copies of \(M\) is CS. \(\Sigma\)-CS modules are direct sums of uniform modules and, more generally, it was also known that every \(\aleph_1\)-\(\Sigma\)-CS module is a direct sum of uniforms, but the question whether every \(\aleph_1\)-\(\Sigma\)-CS module is already \(\Sigma\)-CS remained open. In this paper, the author gives a positive solution to this problem. In order to obtain this result, the author proves the following interesting fact: If \(M\) is a uniform module such that each local direct summand of a countable direct sum of copies of \(M\) is a direct summand, then \(M\) has local endomorphism ring.
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 2104460
zbMATH Keywords: CS modules
zbMATH Keywords: \(\Sigma\)-CS-modules
zbMATH Keywords: indecomposable decompositions
zbMATH Keywords: uniform modules
zbMATH Keywords: local direct summands
zbMATH Keywords: local endomorphism rings
author: Pedro A. Guil Asensio
reviewed by: José L. Gómez-Pardo
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.01.008
OpenAlex ID: W1980170556
cites work: On σ-extending modules
cites work: Q4056148
cites work: RINGS IN WHICH EVERY COMPLEMENT RIGHT IDEAL IS A DIRECT SUMMAND
cites work: WHEN SELF-INJECTIVE RINGS ARE QF: A REPORT ON A PROBLEM
cites work: On indecomposable decompositions of \(CS\)-modules. II
cites work: σ-cs-modules
cites work: Big direct sums of copies of a module have well behaved indecomposable decompositions
cites work: Q4519445
cites work: Every \(\Sigma\)-CS-module has an indecomposable decomposition
cites work: Indecomposable decompositions of modules whose direct sums are CS.
cites work: Rings characterized by direct sums of cs modules
cites work: Q3995216
cites work: Q4002547
cites work: Classes of modules with the exchange property
DOI: 10.1016/J.JALGEBRA.2004.01.008
description: scientific article; zbMATH DE number 2104460