On the existence of covers by injective modules relative to a torsion theory

Abstract

Let R be a ring with identity. In this note we study covers of left R-modules by r-injectives left R-modules, where r is a hereditary torsion theory defined in the category of all left R-modules and all R-morphisms. When R is an artinian commutative ring, a complete answer about the existence of such covers for every R-module is given. In case that T is a centrally splitting torsion theory, we can characterize those T for which every left R-module has a T-injective cover. Also we analyze R-modules such that the injective and the T-injective cover are the same. At the end of this note we relate the concepts of colocalization and cover