A theorem on direct products of slender modules
type: Item
instance of: scholarly article
title: A theorem on direct products of slender modules
zbMATH Open document ID: 0641.16029
author: John D. O'Neill
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108081
review text: R is an associative ring with 1. R-modules X and Y are of the same type if \(Hom_ R(X,Y)\neq 0\neq Hom_ R(Y,X)\). A class \({\mathcal C}\) of R-modules is transitive if, for every X, Y, Z in \({\mathcal C}\), \(Hom_ R(X,Y)\neq 0\neq Hom_ R(Y,Z)\) implies \(Hom_ R(X,Z)\neq 0\). The main result of this paper is as follows: Suppose that \({\mathcal C}\) is a transitive class of slender R-modules with the following property (p): for a countable family \(\{G_ i\}\), \(i\in I\) of modules of the same type, if \(\prod_{i\in I}G_ i=A\oplus B\) then A is isomorphic to the product of members of \({\mathcal C}\). Then the property (p) holds for any family of modules \(G_ i\) in \({\mathcal C}\), for any index set I of non-measurable cardinality.
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 4044036
zbMATH Keywords: direct product
zbMATH Keywords: transitive class of slender R-modules
zbMATH Keywords: modules of the same type
zbMATH Keywords: non-measurable cardinality
MaRDI profile type: Publication
cites work: Q5673086
cites work: On reflexive modules and Abelian groups
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description: scientific article; zbMATH DE number 4044036