A submoduleA of amodule M is said to be strongly pure , if for each finite subset {a i } in A , (equivalently, for each a A) there exists ahomomorphism f : M A such that f(a i ) = a i , i(f(a)=a). A module M is said to be strongly F-regular if each submodule of M is strongly pure . The main purpose of this paper is to develop the properties of strongly F-regular modules and study modules with the property that the intersection of any two strongly pure submodules is strongly pure .

In this paper , we introduce * relation on the Lattic of submodules of a module M. We say that submodule X ,Y of M are * equivalent , X * Y, if +  + and +  + ,for some H≪TM .We show that * relation is an equivalence relation and has good behavior with respect to addition of submodules and homomorphisms. This relation is used to define and study the class of Goldie*-T-lifting modules.