# Some remarks on $u$-$S$-Noetherian and $u$-$S$-coherent rings

## TL;DR

This paper provides new characterizations of $u$-$S$-Noetherian and $u$-$S$-coherent rings using uniform $S$-versions of modules and addresses an open question in the field.

## Contribution

It introduces novel characterizations of these rings via uniform $S$-modules and resolves a previously posed question by Bouziri.

## Key findings

- New characterizations of $u$-$S$-Noetherian rings.
- New characterizations of $u$-$S$-coherent rings.
- Negative answer to Bouziri's question.

## Abstract

In this paper, we give some new characterizations of $u$-$S$-Noetherian rings and $u$-$S$-coherent rings in terms of uniform $S$-version of injective precovers, flat preenvelopes and absolutely pure modules, respectively. Moreover, we give a negative answer to a question proposed by Bouziri [3].