Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The group ${\rm K}_1(\mS_n)$ of the algebra of one-sided inverses of a
  polynomial algebra   The algebra $\mS_n$ of one-sided inverses of a polynomial algebra $P_n$ in
$n$ variables is obtained from $P_n$ by adding commuting, {\em left} (but not
two-sided) inverses of the canonical generators of the algebra $P_n$. The
algebra $\mS_n$ is a noncommutative, non-Noetherian algebra of classical Krull
dimension $2n$ and of global dimension $n$ which is not a domain. If the ground
field $K$ has characteristic zero then the algebra $\mS_n$ is canonically
isomorphic to the algebra $K< \frac{\der}{\der x_1}, ..., frac{\der}{\der x_n},
\int_1, ..., \int_n>$ of scalar integro-differential operators. %Ignoring
non-Noetherian % property, the algebra $\mS_n$ belongs to a family of algebras
% like the $n$th Weyl algebra $A_n$ and the polynomial algebra %$P_{2n}$. It is
proved that $ {\rm K}_1(\mS_n)\simeq K^*$. The main idea is to show that the
group $\GL_\infty (\mS_n)$ is generated by $K^*$, the group of elementary
matrices $E_\infty (\mS_n)$ and $(n-2)2^{n-1}+1$ explicit (tricky) matrices and
then to prove that all the matrices are elementary. For each nonzero idempotent
prime ideal $\gp$ of height $m$ of the algebra $\mS_n$, it is proved that $$
\rm K}_1(\mS_n, \gp)\simeq K^*, & \text{if}m=1, \Z^{\frac{m(m-1)}{2}}\times
K^{*m}& \text{if}m> 1.$$

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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: nlin