A note on Matlis localizations

TL;DR

This paper explores new characterizations of $S$-Matlis rings, focusing on properties related to $S$-strongly flat, $S$-weakly cotorsion, and $S$-$h$-divisible modules, thereby deepening understanding of their structure.

Contribution

It introduces novel characterizations of $S$-Matlis rings using module-theoretic properties, expanding the theoretical framework of these rings.

Findings

New criteria for $S$-Matlis rings based on module properties

Connections between $S$-strongly flat and $S$-weakly cotorsion modules

Characterizations involving $S$-$h$-divisible modules

Abstract

Let $R$ be a commutative ring and $S$ a multiplicative subset of $R$. A ring $R$ is called an $S$-Matlis ring if $pd_RR_S\leq 1$. In this note, we give some new characterizations of $S$-Matlis rings in terms of $S$-strongly flat modules, $S$-weakly cotorsion modules and $S$-$h$-divisible modules.