t 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, Xβ∗∗Y , if and only if XX+Y ⊆ Rad(XM)+X and XY+Y ⊆ Rad(M)+Y 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-Radsupplemented if and only if A and B are Goldie-Rad-supplemented.
