co-Hopfian Modules

TL;DR

This paper investigates co-Hopfian modules over rings, characterizing when finitely generated modules have finite length and analyzing the structure of certain abelian groups with cotorsion torsion subgroups.

Contribution

It provides new characterizations of co-Hopfian modules over commutative Noetherian rings and describes their structure in specific cases.

Findings

Submodules of co-Hopfian injective modules are co-Hopfian under certain conditions.

Finitely generated co-Hopfian modules over these rings have finite length.

Structure of Hopfian and co-Hopfian abelian groups with cotorsion torsion subgroup is described.

Abstract

If $R$ is a ring with 1, we call a unital left $R$-module $M$ co-Hopfian (Hopfian) in the category of left $R$-modules if any monic (epic) endomorphism of $M$ is an automorphism. For commutative Noetherian $R$ we use results of Matlis to show that in a certain context every submodule of a co-Hopfian injective module is co-Hopfian. For these same $R,$ we characterize when a finitely generated co-Hopfian module has finite length. We describe the structure of Hopfian and co-Hopfian abelian groups whose torsion subgroup is cotorsion.