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. 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.

Let R be a ring and M be an R-module. In this paper we investigate modules M such that every (simple) cosingular R-module is M-projective. We prove that every simple cosingular module is Mprojective if and only if for N ≤ T ≤ M , whenever T /N is simple cosingular, then N is a direct summand of T. We show that every simple cosingular right R-module is projective if and only if R is a right GV-ring. It is also shown that for a right perfect ring R, every cosingular right R-module is projective if and only if R is a right GVring. In addition, we prove that if every δ-cosingular right R-module is semisimple, then Z(M) is a direct summand of M for every right R-module M if and only if Z δ (M) is a direct summand of M for every right R-module M .

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The class of H -supplemented modules, which is a nice generalization of that of lifting modules, has been studied extensively in the last decade. As the concept of homomorphisms plays an important role in module theory, we are interested in H -supplemented modules relative to homomorphisms. Let R be a ring, M a right R -module, and S = End R(M ) . We say that M is endomorphism H -supplemented (briefly, E -H -supplemented) provided that for everyIn this paper, we deal with the E -H -supplemented property of modules and also a similar property for a module M by considering Hom R(N, M ) instead of S where N is any module.