Simple-separable modules

Abstract

A module $M$ over a ring is called simple-separable if every simple submodule of $M$ is contained in a finitely generated
 direct summand of $M$. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that
 a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module $M$
 over a right noetherian ring is simple-separable if and only if $M=M_1 \oplus M_2$ such that $M_1$ is separable and $M_2$ has zero socle.
 The structure of simple-separable abelian groups is completely described.