a variation of δ-lifting and δ-supplemented modules with respect to an equivalence relation

In this paper we introduce goldie∗-δ-supplemented modules as follows. a module m is called goldie∗-δ-supplemented (briefly, g∗δ -supplemented) if there exists a δ-supplement t of m for every submodule a of m such that aβδ∗t . we say that a module m is called goldie∗-δ-lifting (briefly, g∗δ -lifting) if there exists a direct summand d of m for every sub- module a of m such that aβδ∗d. note that the last concept given in [4] as a δ-h-supplemented module. we present fundamental properties of these modules. we indicate that these modules lie between δ-lifting and δ-supplemented modules. also we prove that our modules coincide with some variations of δ-supplemented modules for δ-semiperfect modules.