Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity

Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $T$ is called uniformly $S$-torsion provided that $sT=0$ for some $sin S$. The notion of $S$-exact sequences is also introduced from the viewpoint of uniformity. An $R$-module $F$ is called $S$-flat provided that the induced sequence $0rightarrow Aotimes_RFrightarrow Botimes_RFrightarrow Cotimes_RFrightarrow 0$ is $S$-exact for any $S$-exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. A ring $R$ is called $S$-von Neumann regular provided there exists an element $sin S$ satisfies that for any $ain R$ there exists $rin R$ such that $sa=ra^2$. We obtain that a ring $R$ an $S$-von Neumann regular ring if and only if any $R$-module is $S$-flat. Several properties of $S$-flat modules and $S$-von Neumann regular rings are obtained.