Generalized \(\oplus\)-supplemented modules

Abstract

Let \(R\) be a ring and \(M\) be a left \(R\)-module. \(M\) is called generalized \(\oplus\)- supplemented  if every submodule of \(M\) has a generalized supplement that is a direct summand of \(M\). In this paper we give various properties of such modules. We show that any finite direct sum of generalized \(\oplus\)-supplemented modules is generalized \(\oplus\)-supplemented. If \(M\) is a generalized \(\oplus\)-supplemented module with \((D3)\), then every direct summand of \(M\) is generalized \(\oplus\)-supplemented. We also give some properties of generalized cover.