A note on the Cohen type theorem and the Eakin-Nagata type theorem for uniformly $S$-Noetherian rings

TL;DR

This paper extends classical theorems to uniformly S-Noetherian modules and rings, providing new characterizations and solving an open question in the area.

Contribution

It establishes Cohen and Eakin-Nagata type theorems for uniformly S-Noetherian structures and addresses an open problem posed by Kim and Lim.

Findings

Cohen type theorem for uniformly S-Noetherian modules

Eakin-Nagata type theorem for uniformly S-Noetherian rings

Resolution of Kim and Lim's open question

Abstract

In this note, we give the Cohen type theorem for uniformly $S$-Noetherian modules and the Eakin-Nagata type theorem for uniformly $S$-Noetherian rings. We also solve an open question proposed by Kim and Lim [5,Question 4.10].