Abstract

Let $R$ be a commutative ring. It is known that any injective endomorphism of finitely generated $R$-module is an isomorphism if and only if every prime ideal of $R$ is maximal. This result makes possible a characterization of rings on which all finitely generated modules are co-hopfian. The motivation of this paper comes from trying to extend these results to mono-correct modules. In doing so, we show that any finitely generated $R$-module is mono-correct if and only if every prime ideal of $R$ is maximal and we  obtain a characterization of commutative rings on which all finitely generated module are mono-correct. Such rings are exactly commutative strongly $\Pi$-regular rings. So we have a new characterization of commutative strongly $\Pi$-regular rings.

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