A New Characterization of Commutative Strongly $\Pi$-Regular Rings

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 Π-regular rings. So we have a new characterization of commutative strongly Π-regular rings.