$S$-almost perfect commutative rings

TL;DR

This paper characterizes when all modules over a commutative ring have $S$-strongly flat covers, linking it to the perfection of localizations and quotients, even with zero-divisors in $S$.

Contribution

It establishes a precise criterion involving perfect rings for the existence of $S$-strongly flat covers in modules over commutative rings.

Findings

All modules have $S$-strongly flat covers iff all flat modules are $S$-strongly flat.

This occurs iff the localization $R_S$ and all quotients $R/sR$ are perfect rings.

The results hold even when $S$ contains zero-divisors.

Abstract

Given a multiplicative subset $S$ in a commutative ring $R$, we consider $S$-weakly cotorsion and $S$-strongly flat $R$-modules, and show that all $R$-modules have $S$-strongly flat covers if and only if all flat $R$-modules are $S$-strongly flat. These equivalent conditions hold if and only if the localization $R_S$ is a perfect ring and, for every element $s\in S$, the quotient ring $R/sR$ is a perfect ring, too. The multiplicative subset $S\subset R$ is allowed to contain zero-divisors.