Amply (weakly) Goldie-Rad-supplemented modules

Let $R$ be a ring and $M$ be a right $R$-module. We say a submodule $S$ of $M$ is a (weak) Goldie-Rad-supplement of a submodule $N$ in $M$, if $M=N+S$, $(N\cap S \leq Rad(M))$ $N\cap S\leq Rad(S)$ and $N\beta^{**} S$, and $M$ is called amply (weakly) Goldie-Rad-supplemented if every submodule of $M$ has ample (weak) Goldie-Rad-supplements in $M$. In this paper we study various properties of such modules. We show that every distributive projective weakly Goldie-Rad-Supplemented module is amply weakly Goldie-Rad-Supplemented. We also show that if $M$ is amply (weakly) Goldie-Rad-supplemented and satisfies DCC on (weak) Goldie-Rad-supplement submodules and on small submodules, then $M$ is Artinian.