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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 $R$, $x\notin R\implies x^{{1}}\in R$.
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.
Keywords:
Prufer domain
Related:
PruferDomain
Type of Math Object:
Definition
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Reference
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