Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: cs
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph