Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rings whose indecomposable modules are pure-projective or pure-injective   It is proven each ring $R$ for which every indecomposable right module is
pure-projective is right pure-semisimple. Each commutative ring $R$ for which
every indecomposable module is pure-injective is a clean ring and for each
maximal ideal $P$, $R_P$ is a maximal valuation ring. Complete discrete
valuation domain of rank one are examples of non-artinian semi-perfect rings
with pure-injective indecomposable modules.

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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat