Endo-Artinian Modules

Authors

  • Saeed Safaeeyan Yasouj University

Abstract

Let $R$ be a commutative ring. A right $R$-module $M$ is said to be endo-artinian if it is artinian with regard to the left $L$-module structure, where $L={\rm End}_R(M)$. This study demonstrates that if $R$ is a Dedekind domain, then all injective $R$-modules with finitely many simple components, all unfaithful modules, and arbitrary direct sums of an endo-artinian module are endo-artinian modules.  Moreover, the following result is established: if $R$ is a Dedekind domain and $M$ is an indecomposable injective torsion right $R$-module, then $M$ is an artinian module. It can therefore be demonstrated, on the basis of the preceding arguments, that if $S$ is a simple $R$-module, then its injective hull $E(S)$ is an Artinian module. Finally, assuming that the ring $R$ is a Dedekind domain, we will present a necessary and sufficient condition for an $R$-module to be an endo-artinian module.

Published

2025-09-03

Issue

Vol. 19

Section

Vol. 19, No. 4, (2025)

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