On a variation of \(\oplus\)-supplemented modules

Abstract

Let \(R\) be a ring and \(M\) be an \(R\)-module. \(M\) is called \(\oplus_{ss}\)-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 \(\oplus_{ss}\)-supplemented modules are provided. In particular, it is shown that \((1)\) if a module \(M\) is \(\oplus_{ss}\)-supplemented, then \(Rad(M)\) is semisimple and \(Soc(M)\unlhd M\); \((2)\) every direct sum of \(ss\)-lifting modules is \(\oplus_{ss}\)-supplemented; \((3)\) a commutative ring \(R\) is an artinian serial ring with semisimple radical if and only if every left \(R\)-module is \(\oplus_{ss}\)-supplemented.