A variation of \delta-lifting and \delta-supplemented modules with respect to an equivalence relation

In this paper we introduce Goldie^{\ast }-\delta -supplemented modules as follows. A module M is called Goldie^{\ast }-\delta -supplemented (briefly, G_{\delta }^{\ast }-supplemented) if there exists a \delta -supplement T of M for every submodule A of M such that A\beta_{\delta }^{\ast }T. We say that a module M is called Goldie^{\ast }-\delta -lifting (briefly, G_{\delta }^{\ast }-lifting) if there exists a direct summand D of M for every submodule A of M such that A\beta_{\delta }^{\ast }D. Note that the last concept given in [۴] as a \delta -H-supplemented module. We present fundamental properties of these modules. We indicate that these modules lie between \delta -lifting and \delta -supplemented modules. Also we prove that our modules coincide with some variations of \delta -supplemented modules for \delta -semiperfect modules.