Uniformly $S$-Noetherian rings

TL;DR

This paper introduces the concept of uniformly $S$-Noetherian rings, explores their properties, extends classical theorems to this setting, and studies related modules and ring constructions.

Contribution

It defines uniformly $S$-Noetherian rings, proves an adapted Eakin-Nagata-Formanek theorem, and establishes the Cartan-Eilenberg-Bass theorem for these rings.

Findings

Established the Eakin-Nagata-Formanek theorem for $u$-$S$-Noetherian rings.

Analyzed $u$-$S$-Noetherian properties in various ring constructions.

Developed the theory of $u$-$S$-injective modules and proved related theorems.

Abstract

Let $R$ be a ring and $S$ a multiplicative subset of $R$. Then $R$ is called a uniformly $S$-Noetherian ($u$-$S$-Noetherian for abbreviation) ring provided there exists an element $s\in S$ such that for any ideal $I$ of $R$, $sI \subseteq K$ for some finitely generated sub-ideal $K$ of $I$. We give the Eakin-Nagata-Formanek Theorem for $u$-$S$-Noetherian rings. Besides, the $u$-$S$-Noetherian properties on several ring constructions are given. The notion of $u$-$S$-injective modules is also introduced and studied. Finally, we obtain the Cartan-Eilenberg-Bass Theorem for uniformly $S$-Noetherian rings.