On ⊕-δss-supplemented modules

Abstract

A module M is called ?-?ss-supplemented if every submodule X of M has a ?ss-supplement Y in M which is a direct summand of M such that X + Y = M and X ? Y ? Soc? (Y) where Soc?(Y) is the sum of simple and ?-small submodules of Y and M = Y ? Y? for some Y? ? M. Moreover, M is called a completely ?-?ss-supplemented module if every direct summand of M is ?-?ss-supplemented. Thus, we present two new types of algebraic structures which are stronger than ?-D11 and ?-D+11-modules, respectively. In this paper we investigate basic properties, decompositions and ring characterizations of these modules.