# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online May 11, 2022

Copyright © The Korean Mathematical Society.

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

Xiaolei Zhang

Chengdu Aeronautic Polytechnic

### Abstract

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 $s\in S$. 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 $0\rightarrow A\otimes_RF\rightarrow B\otimes_RF\rightarrow C\otimes_RF\rightarrow 0$ is $S$-exact for any $S$-exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. A ring $R$ is called $S$-von Neumann regular provided there exists an element $s\in S$ satisfies that for any $a\in R$ there exists $r\in R$ 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.

Keywords: uniformly $S$-torsion modules, $S$-exact sequences, $S$-flat modules, $S$-von Neumann regular

MSC numbers: 13C12, 16D40, 16E50.