Let $ R $ be a ring. A morphism $\alpha: X\rightarrow Y $ of left $ R $-modules is called a
finitely phantom morphism if for each morphism ${\beta: F\rightarrow X} $ with $ F $ finitely
generated, the composition $\alpha\beta $ factors through a projective left $ R $-module. An
epimorphism $ M\rightarrow N $ of left $ R $-modules is called finitely split if ${\rm Hom} _
{R}(F, M)\rightarrow {\rm 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 …