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arXiv · 2112.01596

Hopfian and co-Hopfian modules over Artinian rings

F. C. Leary
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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.

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F. C. Leary. 2021-12-02. Hopfian and co-Hopfian modules over Artinian rings. https://arxiv.org/abs/2112.01596

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