A note on generalizations of semisimple modules

A left module M over an arbitrary ring is called an RD-module (or an RS-module) if every submodule N of M with Rad(M) subset of N is a direct summand of (a supplement in, respectively) M. In this paper, we investigate the various properties of RD-modules and RS-modules. We prove that M is an RD-module if and only if M = Rad(M) circle plus X, where X is semisimple. We show that a finitely generated RS-module is semisimple. This gives us the characterization of semisimple rings in terms of RS-modules. We completely determine the structure of these modules over Dedekind domains.