valuation domain
An integral domain^{} $R$ is a valuation domain if for all $a,b\in R$, either $ab$ or $ba$. Equivalently, an integral domain is a valuation domain if for any $x$ in the field of fractions^{} of $R$, $x\notin R\u27f9{x}^{1}\in R$.
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^{}.
Title  valuation domain 

Canonical name  ValuationDomain 
Date of creation  20130322 13:47:31 
Last modified on  20130322 13:47:31 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  7 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 16U10 
Classification  msc 13G05 
Classification  msc 13F30 
Related topic  PruferDomain 