Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On countably $\Sigma$-C2 rings   Let $R$ be a ring. $R$ is called a right countably $\Sigma$-C2 ring if every
countable direct sum copies of $R_{R}$ is a C2 module. The following are
equivalent for a ring $R$: (1) $R$ is a right countably $\Sigma$-C2 ring. (2)
The column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\mathbb{N}}(R)$
is a right C2 (or C3) ring. (3) Every countable direct sum copies of $R_{R}$ is
a C3 module. (4) Every projective right $R$-module is a C2 (or C3) module. (5)
$R$ is a right perfect ring and every finite direct sum copies of $R_{R}$ is a
C2 (or C3) module. This shows that right countably $\Sigma$-C2 rings are just
the rings whose right finitistic projective dimension
r$FPD(R)$=sup\{$Pd_{R}(M)|$ $M$ is a right $R$-module with
$Pd_{R}(M)<\infty$\}=0, which were introduced by Hyman Bass in 1960.

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The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
3
The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat
1
The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: econ