ON COMMUTATIVE PFGI-RINGS

Let $R$ be a ring. An $R$-module $M$ is called purely co-Hopfian if for any injective endomorphism $f:M \to M$, $Imf$ is pure in $M$, i.e., $IM \cap Imf = IImf$ for any ideal $I$ of $R$. Let $F_R$ be the class of finitely generated $R$-modules and $P_R$ the class of purely co-Hopfian $R$-modules. In [1] Vasconcelos proved that the class of finitely generated $R$‑modules is included in the class of co-Hopfian $R$-modules if $R$ is a commutative ring whose all prime ideals are maximal. Hence from the transitivity of inclusion, we have: $F_R \subseteq P_R$. In this article, we characterize commutative rings $R$ on which an $R$‑module $M$ is purely co-Hopfian if and only if $M$ is finitely generated; i.e., $P_R=F_R$.