  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.
