$S$-flat cotorsion pair
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Let $R$ be a commutative ring, and let $S$ be a multiplicative subset of $R$. In this paper, we investigate the notion of $S$-cotorsion modules. An $R$-module $C$ is called $S$-cotorsion if $\Ext^{1}_{R}(F,C) = 0$ for every $S$-flat $R$-module $F$. Among other results, we establish that the pair $(S\mathcal{F}, S\mathcal{C})$, where $S\mathcal{F}$ denotes the class of all $S$-flat $R$-modules and $S\mathcal{C}$ denotes the class of all $S$-cotorsion modules, forms a hereditary perfect cotorsion pair. As applications, we provide characterizations of $S$-perfect rings in terms of $S$-cotorsion modules. We conclude the paper with results on $S\mathcal{F}$-preenvelopes. Namely, we prove that if every module has an $S\mathcal{F}$-preenvelope, then $R$ is $S$-coherent. Furthermore, we establish the converse under the condition that $R_S$ is a finitely presented $R$-module.
Abstract
Let $R$ be a commutative ring, and let $S$ be a multiplicative subset of $R$. In this paper, we investigate the notion of $S$-cotorsion modules. An $R$-module $C$ is called $S$-cotorsion if $\Ext^{1}_{R}(F,C) = 0$ for every $S$-flat $R$-module $F$. Among other results, we establish that the pair $(S\mathcal{F}, S\mathcal{C})$, where $S\mathcal{F}$ denotes the class of all $S$-flat $R$-modules and $S\mathcal{C}$ denotes the class of all $S$-cotorsion modules, forms a hereditary perfect cotorsion pair. As applications, we provide characterizations of $S$-perfect rings in terms of $S$-cotorsion modules. We conclude the paper with results on $S\mathcal{F}$-preenvelopes. Namely, we prove that if every module has an $S\mathcal{F}$-preenvelope, then $R$ is $S$-coherent. Furthermore, we establish the converse under the condition that $R_S$ is a finitely presented $R$-module.