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Abstract

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 split epimorphisms. Some applications are also given.