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

In this paper, we introduce the concept of modules with the properties (RE) and (SRE), and we provide various properties of these modules. In particular, we prove that a semisimple module $M$ is $\operatorname{Rad}$-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).