ON THE EXISTENCE OF COVERS BY INJECTIVE-MODULES RELATIVE TO A TORSIONTHEORY

Let R be a ring with identity. In this note we study covers of left R- modules by tau-injectives left R-modules, where tau is a hereditary to rsion theory defined in the category of all left R-modules and all R-m orphisms. When R is an artinian commutative ring, a complete answer ab out the existence of such covers for every R-module is given. In case that tau is a centrally splitting torsion theory, we can characterize those tau for which every left R-module has a tau-injective cover. Als o we analyze R-modules such that the injective and the tau-injective c over are the same. At the end of this note we relate the concepts of c olocalization and cover.