Hopfian and co-Hopfian modules over Artinian rings

Abstract

An $R$-module $M$ is Hopfian (co-Hopfian) if any epic (monic) endomorphism of $M$ is an automorphism. If $R$ is commutative Noetherian, we characterize the co-Hopfian injective $R$-modules, and the Hopfian injectives in the case that $R$ is also reduced. For a commutative Artinian principal ideal ring, we show that $M$ is Hopfian (co-Hopfian) if and only if $M$ is finitely generated if and only if its injective envelope $E(M)$ is Hopfian (co-Hopfian) if and only if $E(M)$ is finitely generated. We identify the obstacle to generalizing this result to arbitrary Artinian principal ideal rings.