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 $u$-$S$-torsion ($u$- always abbreviates uniformly) provided that $sT=0$ for some $s\in S$. The notion of $u$-$S$-exact sequences is also introduced from the viewpoint of uniformity. An $R$-module $F$ is called $u$-$S$-flat provided that the induced sequence $0\rightarrow A\otimes_RF\rightarrow B\otimes_RF\rightarrow C\otimes_RF\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. A ring $R$ is called $u$-$S$-von Neumann regular provided there exists an element $s\in S$ satisfying that for any $a\in R$ there exists $r\in R$ such that $sa=ra^2$. We obtain that a ring $R$ is a $u$-$S$-von Neumann regular ring if and only if any $R$-module is $u$-$S$-flat. Several properties of $u$-$S$-flat modules and $u$-$S$-von Neumann regular rings are obtained.

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