Modules which have a rad-supplement that is a direct summand in every extension

Abstract

In this paper, we introduce the concept of mod- ules with the properties (RE) and (SRE), and we provide various

properties of these modules. In particular, we prove that a semisimple module M is Rad-supplementing if and only if M has the property (SRE). Moreover, we show that a ring R is a left V-ring if and only if every left R-module with the property (RE) is injective. Finally, we characterize the rings whose modules have the properties (RE) and (SRE).