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 $R$ , $x\notin R\implies x^{-1}\in R$ .

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.