New Characterizations of $S$-coherent rings

TL;DR

This paper introduces the class of $S$-Mittag-Leffler modules relative to flat modules, characterizes $S$-coherent rings via module closure properties, and extends classical theorems to the $S$-context, answering an open question.

Contribution

It defines and studies $S$-$\mathcal{F}$-ML modules, providing new characterizations of $S$-coherent rings and extending the Chase Theorem to the $S$-setting.

Findings

$S$-$\mathcal{F}$-ML modules are characterized by closure under submodules.

A ring is $S$-coherent iff $S$-$\mathcal{F}$-ML is closed under submodules.

The $S$-version of Chase Theorem links $S$-coherence with $S$-flatness of direct products.

Abstract

In this paper, we introduce and study the class $S$-$\mathcal{F}$-ML of $S$-Mittag-Leffler modules with respect to all flat modules. We show that a ring $R$ is $S$-coherent if and only if $S$-$\mathcal{F}$-ML is closed under submodules. As an application, we obtain the $S$-version of Chase Theorem: a ring $R$ is $S$-coherent if and only if any direct product of $R$ is $S$-flat if and only if any direct product of flat $R$-modules is $S$-flat. Consequently, we provide an answer to the open question proposed by D. Bennis and M. El Hajoui [3].