$\mathcal{F_{S}}$-Mittag-Leffler modules and global dimension relative to $\mathcal{F_{S}}$-Mittag-Leffler modules

Bull. Korean Math. Soc. Published online 2019 May 16

Mingzhao Chen and Fanggui Wang
Sichuan Normal University

Abstract : Let $R$ be any commutative ring and $S$ be any multiplicative closed set. We use an $S$-version to generalize $\mathcal{F}$-Mittag-Leffler modules, called $\mathcal{F_{S}}$-Mittag-Leffler modules, and define the projective dimension with respect to these modules. We give some characterizations of $\mathcal{F_{S}}$-Mittag-Leffler modules, investigate the relationships between $\mathcal{F}$-Mittag-Leffler modules and $\mathcal{F_{S}}$-Mittag-Leffler modules, and use these relations to describe noetherian rings and coherent rings, such as $R$ is noetherian if and only if $R_{S}$ is noetherian and every $\mathcal{F_{S}}$-Mittag-Leffler module is $\mathcal{F}$-Mittag-Leffler. Besides, We also investigate the $\mathcal{M}^\mathcal{F_{S}}$-global dimension of $R$, and prove that $R_{S}$ is noetherian if and only if its $\mathcal{M}^\mathcal{F_{S}}$-global dimension is zero; $R_{S}$ is coherent if and only if its $\mathcal{M}^\mathcal{F_{S}}$-global dimension is at most one.