FINITELY PHANTOM MORPHISMS AND FINITELY SPLIT EPIMORPHISMS

Let R be a ring. A morphism alpha : X -> Y of left R-modules is called a finitely phantom morphism if for each morphism beta : F -> X with F finitely generated, the composition cxfi factors through a projective left R-module. An epimorphism M -> N of left R-modules is called finitely split if Hom(R)(F, M) -> Hom(R)(F, N) is an epimorphism for any finitely generated left R-module F. We obtain many properties of finitely phantom morphisms and finitely split epimorphisms. Some applications are also given.