Goldie-Rad-Supplemented Modules

In this paper we introduce β** relation on the lattice of submodules of a module M. We say that submodules X, Y of M are β** equivalent, X β** Y, if and only if X+YX⊆Rad(M)+XX${{X + Y} \over X} \subseteq {{Rad(M) + X} \over X}$ and X+YY⊆Rad(M)+YY${{X + Y} \over Y} \subseteq {{Rad(M) + Y} \over Y}$ . We show that the β** 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 ⊕ B is Goldie-Rad-supplemented if and only if A and B are Goldie-Rad-supplemented.