Radicals and Preradicals in the Category of Modules over All Rings

Abstract

Let Mod be a category whose objects are all possible pairs (A, U), where U is an associative ring, A is a right U-module (not unitary in the general case) and the set of morphisms of module (A, U) to module (B, V) consists of pairs of mappings (ϕA, ϕU), where ϕA or ϕU, respectively, is a homomorphism of Abelian group A to Abelian group B (ring U to ring V), where (a ∙ u) ϕA = a ϕA ∙ u ϕU, a ϵ A, u ϵ U. This pair of mappings is called a homomorphism of module (A, U) to module (B, V). It is proved that strict radicals of Mod in the sense of Kurosh are described by means of systems of strict radicals of the category of associative rings As and categories of right U-modules Mod — U, U ϵ As. It turned out that a wider classes of preradicals and radicals of Mod can also be described by means of systems of preradicals and radicals of As and Mod — U, U ϵ As, respectively, which satisfy some conditions.