A theorem on direct products of slender modules
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.
scientific article; zbMATH DE number 4044036
scientific article