valuation domainValuationDomain2003-07-25 18:22:402004-11-30 21:47:16DefinitionPrufer domain% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}An integral domain $R$ is a \emph{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 $R$, $x\notin R\implies x^{-1}\in R$.
Some properties:
\begin{itemize}
\item 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.
\item 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.
\item Valuation domains are integrally closed.
\end{itemize}