In this paper we introduce relation on the lattice of submod- ules of a module M. We say that submodules X; Y of M are equivalent, X Y , if and only if X+Y X Rad(M)+X X and X+Y Y Rad(M)+Y Y . We show that the relation is an equivalence relation. We also investigate some general properties of this relation. This rela- tion is used to de ne 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.
