Let \(R\) be a commutative ring and let \(Q\) be its total ring of quotients. An \(R\)-module \(C\) is said to be weakly cotorsion if \(\mathrm{Ext}^1_R(Q, C) = 0\). An \(R\)-module \(F\) is strongly flat if \(\mathrm{Ext}^1_ R(F, C) = 0\) for all weakly cotorsion \(R\)-modules \(C\). The two present authors are among the people who studied these two notions for several years and have published their results in several papers.
In the paper under review, the authors consider \(S\)-weakly cotorsion and \(S\)-strongly flat \(R\)-modules for a given multiplicative subset \(S\) in a commutative ring \(R\), 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. Note that the multiplicative subset \(S\subset R\) is allowed to contain zero-divisors.
The paper is well-written with all information that one need to read.