A theorem on direct products of slender modules. (English) Zbl 0641.16029

O’Neill, John D.

Rend. Semin. Mat. Univ. Padova 78, 261-266 (1987).

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.

Reviewer: R.M.Dimitrić

MSC:

16Gxx	Representation theory of associative rings and algebras
16D80	Other classes of modules and ideals in associative algebras
20K25	Direct sums, direct products, etc. for abelian groups
16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
13C13	Other special types of modules and ideals in commutative rings

Keywords:

direct product; transitive class of slender R-modules; modules of the same type; non-measurable cardinality

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