Generalized $\oplus$-supplemented modules

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.