Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Invertibility of Matrices over Subrings   Given rings $R \subseteq S$, consider the division closure $DC(R,S)$ and the
rational closure $RC(R,S)$ of R in S. If S is commutative, then
$DC(R,S)=RC(R,S)=RT^{-1}$, where $T = \{t\in R : t^{-1} \in S\}$. We show that
this is also true if we assume only that R is commutative.

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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc