Hopfian and co-Hopfian modules over Artinian rings

TL;DR

This paper characterizes Hopfian and co-Hopfian modules over Artinian rings, especially focusing on injective modules over commutative Noetherian and Artinian principal ideal rings, revealing conditions for their finiteness and automorphism properties.

Contribution

It provides new characterizations of Hopfian and co-Hopfian modules over Artinian rings, including conditions relating modules and their injective envelopes, and discusses limitations in generalizing these results.

Findings

Characterization of co-Hopfian injective modules over commutative Noetherian rings.

Equivalence of Hopfian/co-Hopfian property and finite generation for modules over Artinian principal ideal rings.

Identification of obstacles in extending results to arbitrary Artinian principal ideal 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.