An extension of $S$-noetherian rings and modules

TL;DR

This paper generalizes the concept of $S$-noetherian rings by using hereditary torsion theories, establishing properties like finiteness of type and locality for totally $\sigma$-noetherian rings.

Contribution

It introduces a new framework for $S$-noetherian rings via hereditary torsion theories, extending classical notions and proving key properties.

Findings

Hereditary torsion theory $\sigma$ of finite type for totally $\sigma$-noetherian rings

Totally $\sigma$-noetherian property is local

Generalization encompasses classical $S$-noetherian rings

Abstract

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