Generalized ⊕-supplemented modules

Abstract

Let R be a ring and M be a left R-module. M is called generalized ⊕- 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 ⊕-supplemented modules is generalized ⊕-supplemented. If M is a generalized ⊕-supplemented module with (D3), then every direct summand of M is generalized ⊕-supplemented. We also give some properties of generalized cover.