On Direct Sums of Injective Modules and Chain Conditions

Abstract

Abstract Let R be a ring and M a right R -module. Let σ[ M ] be the full subcategory of Mod- R subgenerated by M . An M -natural class 𝒦 is a subclass of σ[ M ] closed under submodules, direct sums, isomorphic copies, and M -injective hulls. We present some equivalent conditions each of which describes when σ has the property that direct sums of ( M -)injective modules in σ are ( M -)injective. Specializing to particular M , and/or special subclasses we obtain many new results and known results as corollaries.