Goldie-Rad-Supplemented Modules

Abstract

In this paper we introduce beta** relation on the lattice of submodules of a module M. We say that submodules X, Y of M are beta** equivalent, X beta**Y, if and only if X+Y/X subset of Rad(M)+X/X and X+Y/Y subset of Rad(M)+Y/Y We show that the beta** relation is an equivalence relation. We also investigate some general properties of this relation. This relation is used to define and study classes of Goldie-Rad-supplemented and Rad-H-supplemented modules. We prove M = A circle plus B is Goldie-Rad-supplemented if and only if A and B are Goldie-Rad-supplemented.