A note on $S$-flat preenvelopes

TL;DR

This paper explores the properties of S-flat preenvelopes in module theory, providing counterexamples that challenge assumptions about coherence and S-coherence in rings.

Contribution

It demonstrates that coherence of a ring does not guarantee the existence of S-flat preenvelopes for all modules, and that S-coherence does not imply ring coherence.

Findings

A ring being coherent does not ensure all modules have S-flat preenvelopes.

R_S being coherent does not imply R is S-coherent.

Counterexamples to common assumptions in module and ring theory.

Abstract

In this note, we investigate the notion of $S$-flat preenvelopes of modules. In particular, we give an example that a ring $R$ being coherent does not imply that every $R$-module have an $S$-flat preenvelope, giving a negative answer to the question proposed by Bennis and Bouziri \cite{BB24}. Besides, we also show that $R_S$ is a coherent ring also does not imply that $R$ is an $S$-coherent ring in general.