Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $\star $-Power Conductor domains   Let $D$ be an integral domain and $\star $ a star operation defined on $D$.
We say that $D$ is a $\star $-power conductor domain ($\star $-PCD) if for each
pair $a,b\in D\backslash (0)$ and for each positive integer $n$ we have
$Da^{n}\cap Db^{n}=((Da\cap Db)^{n})^{\ast }.$ We study $\star $-PCDs and
characterize them as root closed domains satisfying $
((a,b)^{n})^{-1}=(((a,b)^{-1})^{n})^{\star }$ for all nonzero $a,b$ and all
natural numbers $n\geq 1$. From this it follows easily that Pr\"{u}fer domains
are $d$-PCDs (where $d$ denotes the trivial star operation), and $v$ -domains
(e.g., Krull domains) are $v$-PCDs, thereby establishing that a $v$ -domain
(e.g., a Prufer or Krull domain) is a $\star $ -PCD. We also consider when a
$\star $-PCD is completely integrally closed, and this leads to new
characterizations of Krulll domains. In particular, we show that a Noetherian
domain is a Krull domain if and only if it is a $w$ -PCD.

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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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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: cmp-lg
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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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: cs