A note on $S$-coherent rings

TL;DR

This paper characterizes $S$-coherent rings by their finitely presented modules and explores the relationship between $c$-$S$-coherent and $S$-coherent rings, providing examples and answering a previously posed question.

Contribution

It proves that a ring is $S$-coherent if and only if all finitely presented modules are $S$-coherent, and clarifies the relationship between $c$-$S$-coherent and $S$-coherent rings.

Findings

A ring is $S$-coherent iff all finitely presented modules are $S$-coherent.

$c$-$S$-coherent rings are $S$-coherent.

Counterexample showing the converse does not hold.

Abstract

In this note, we show that a ring $R$ is $S$-coherent if and only if every finitely presented $R$-module is $S$-coherent, providing a positive answer to a question proposed in [D. Bennis, M. El Hajoui, {\it On $S$-coherence}, J. Korean Math. Soc. \textbf{55} (2018), no.6, 1499-1512]. Besides, we show that $c$-$S$-coherent rings are $S$-coherent, and give an example to show the converse is not true in general.