On a variation of ⊕-supplemented modules

Abstract

Let R be a ring and M be an R-module. M is called circle plus(ss)-supplemented if every submodule of M has a ss-supplement that is a direct summand of M. In this paper, the basic properties and characterizations of circle plus(ss)-supplemented modules are provided. In particular, it is shown that (1) if a module M is circle plus(ss)-supplemented, then Rad(M) is semisimple and Soc(M) (sic) M; (2) every direct sum of ss-lifting modules is circle plus(ss)-supplemented; (3) a commutative ring R is an artinian serial ring with semisimple radical if and only if every left R-module is circle plus(ss)-supplemented.