Strongly flat modules via universal localization

In this paper, we investigate a non-commutative version of strongly flat modules, which is based on the concept of universal localization introduced by Cohn. We consider a set $\sigma$ consisting of maps of finitely generated projective $R$-modules, where $R$ is not necessarily a commutative ring. Let $R_{\sigma }$ denote the universal localization of $R$ with respect to $\sigma$. The class of $\sigma$-strongly flat modules is defined as the left class in the cotorsion pair generated by $R_{\sigma }$. We examine the homotopy category of $\sigma$-strongly flat modules and demonstrate that the thick subcategory ${\mathcal{S}}_{\sigma }$, consisting of acyclic complexes, wherein all syzygies are $\sigma$-strongly flat, forms a precovering class within this homotopy category. This implies that the quotient map from $\mathbb{K}(\sigma -\mathcal{S}\mathcal{F})$ to $\mathbb{K}(\sigma -\mathcal{S}\mathcal{F})\mathrm{/}{\mathcal{S}}_{\sigma }$ always has a fully faithful right adjoint.