Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Strong cleanness of the $2\times 2$ matrix ring over a general local
  ring   A ring $R$ is called strongly clean if every element of $R$ is the sum of a
unit and an idempotent that commute with each other. A recent result of
Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the
commutative local rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For
a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring
${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local
rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.

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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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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: chem-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: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: nlin