Stable torsion theories and the injective hulls of simple modules

TL;DR

This paper characterizes when injective hulls of simple modules over left Noetherian rings are locally Artinian using torsion theories, providing new conditions for this finiteness property.

Contribution

It introduces a torsion theoretical framework to characterize and identify rings with locally Artinian injective hulls of simple modules, advancing understanding of module finiteness conditions.

Findings

Characterization of rings with locally Artinian injective hulls via torsion theories

Sufficient conditions for Noetherian rings to have this property

New torsion-theoretic criteria for module finiteness

Abstract

A torsion theoretical characterization of left Noetherian rings $R$ over which injective hulls of simple left modules are locally Artinian is given. Sufficient conditions for a left Noetherian ring to satisfy this finiteness condition are obtained in terms of torsion theories.