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 $R$, $x\notin R⟹x^{-1}\in R$.

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.

Title	valuation domain
Canonical name	ValuationDomain
Date of creation	2013-03-22 13:47:31
Last modified on	2013-03-22 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