Amply (weakly) Goldie-Rad-supplemented modules

Abstract

Let \(R\) be a ring and \(M\) be a right \(R\)-module. We say a submodule \(S\) of \(M\) is a \textit{(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.