Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unique factorization property of non-unique factorization domains II   Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a
valuation element if there is a valuation overring $V$ of $D$ such that $aV\cap
D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each
nonzero nonunit of $D$ can be written as a finite product of valuation
elements. In this paper, we study some ring-theoretic properties of VFDs. Among
other things, we show that (i) a VFD $D$ is Schreier, and hence ${\rm
Cl}_t(D)=\{0\}$, (ii) if $D$ is a P$v$MD, then $D$ is a VFD if and only if $D$
is a weakly Matlis GCD-domain, if and only if $D[X]$, the polynomial ring over
$D$, is a VFD and (iii) a VFD $D$ is a weakly factorial GCD-domain if and only
if $D$ is archimedean. We also study a unique factorization property of VFDs.

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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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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: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph