A note on Matlis localizations

Let $R$ be a commutative ring and $S$ a multiplicative subset of $R$. A ring $R$ is called an $S$-Matlis ring if $p{d}_R{R}_S\le 1$. 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.