Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2022; 59(3): 643-657

Online first article May 11, 2022      Printed May 31, 2022

https://doi.org/10.4134/BKMS.b210291

Copyright © The Korean Mathematical Society.

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

Xiaolei Zhang

Shandong University of Technology

Abstract

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.

Keywords: $u$-$S$-torsion module, $u$-$S$-exact sequence, $u$-$S$-flat module, $u$-$S$-von Neumann regular ring

MSC numbers: 13C12, 16D40, 16E50

Supported by: The author was supported by the National Natural Science Foundation of China (No. 12061001).

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