On S-coherence

TL;DR

This paper introduces $S$-coherent rings and $S$-finitely presented modules, extending classical concepts with localization techniques, and provides characterizations and discussions on these $S$-versions.

Contribution

It defines $S$-coherent rings and $S$-finitely presented modules, offering new characterizations and extending classical results to the $S$-context.

Findings

Introduces $S$-coherent rings and $S$-finitely presented modules.

Provides an $S$-version of Chase's characterization of coherent rings.

Characterizes $S$-versions via localization.

Abstract

Recentely, Anderson and Dumitrescu's $S$-finiteness has attracted the interest of several authors. In this paper, we introduce the notions of $S$-finitely presented modules and then of $S$-coherent rings which are $S$-versions of finitely presented modules and coherent rings, respectively. Among other results, we give an $S$-version of the classical Chase's characterization of coherent rings. We end the paper with a brief discussion on other $S$-versions of finitely presented modules and coherent rings. We prove that these last $S$-versions can be characterized in terms of localization.