Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonsplit module extensions over the one-sided inverse of k[x]   Let $R$ be the associative $k$-algebra generated by two elements $x$ and $y$
with defining relation $yx=1$. A complete description of simple modules over
$R$ is obtained by using the results of Irving and Gerritzen. We examine the
short exact sequence $0\rightarrow U\rightarrow E \rightarrow V\rightarrow 0$,
where $U$ and $V$ are simple $R$-modules. It shows that nonsplit extension only
occurs when both $U$ and $V$ are one-dimensional, or, under certain condition,
$U$ is infinite-dimensional and $V$ is one-dimensional.

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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph