Modules Whose Endomorphism Rings are Unit-Regular

Unit-regular rings have been extensively studied in the literature. In this article, we introduce the notion of unit endoregular modules which generalizes that of unit-regular rings. A module is called unit endoregular if its endomorphism ring is unit-regular. It is proved that a ring R is a right V-ring if and only if every finitely cogenerated right R-module is unit endoregular. Some characterizations of unit endoregular modules are provided so that several known results on unit-regular rings are generalized. We also provide some properties of unit endoregular modules. For instance, any module which is mutually subisomorphic to a unit endoregular module M is isomorphic to M.