An extension of S-artinian rings and modules to a hereditary torsion theory setting

TL;DR

This paper generalizes the concept of S-artinian rings to a hereditary torsion theory context, establishing conditions under which such rings are also noetherian, thereby broadening the understanding of ring and module structures.

Contribution

It introduces a new framework for S-artinian rings using hereditary torsion theories, extending classical notions and proving key properties of totally σ-artinian rings.

Findings

If A is totally σ-artinian, then σ is of finite type.

Totally σ-artinian rings are also totally σ-noetherian.

The generalization links torsion theories with classical ring properties.

Abstract

For any commutative ring $A$ we introduce a generalization of $S$--artinian rings using a hereditary torsion theory $\sigma$ instead of a multiplicative closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $\sigma$--artinian ring, then $\sigma$ must be of finite type, and $A$ is totally $\sigma$--noetherian.