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 (weak) Goldie-Rad-supplement of a submodule N in M, if M = N + S, (N boolean AND S <= Rad(M)) N boolean AND S <= Rad(S) and N beta**S, andM is called am ply (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.