valuation domain

An integral domain $R$ is a valuation domain if for all $a,b\\in R$ , either $a|b$ or $b|a$ . Equivalently, an integral domain is a valuation domain if for any $x$ in the field of fractions of $R$ , $x\otin R\\implies x^{-1}\\in R$ .

Some properties:

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A valuation domain is a discrete valuation ring (DVR) if and only if it is a principal ideal domain (PID) if and only if it is Noetherian.
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Every valuation domain is a Bezout domain, though the converse is not true. For a partial converse, any local Bezout domain is a valuation domain.
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Valuation domains are integrally closed.

单词	ValuationDomain
释义	valuation domain An integral domain $R$ is a valuation domain if for all $a,b\\in R$ , either $a\|b$ or $b\|a$ . Equivalently, an integral domain is a valuation domain if for any $x$ in the field of fractions of $R$ , $x\otin R\\implies x^{-1}\\in R$ . Some properties: • A valuation domain is a discrete valuation ring (DVR) if and only if it is a principal ideal domain (PID) if and only if it is Noetherian. • Every valuation domain is a Bezout domain, though the converse is not true. For a partial converse, any local Bezout domain is a valuation domain. • Valuation domains are integrally closed.
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