PlanetMath: valuation domain

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

valuation domain

(Definition)

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

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.

"valuation domain" is owned by mathcam.

(view preamble)

See Also: Prüfer domain

Keywords:

Prufer domain

Attachments:: valuation domain is local (Theorem) by pahio
place of field (Theorem) by pahio

Log in to rate this entry.
(view current ratings)

Cross-references: integrally closed, converse, Bezout domain, Noetherian, principal ideal domain, discrete valuation ring, properties, field of fractions, integral domain
There are 6 references to this entry.

This is version 4 of valuation domain, born on 2003-07-25, modified 2004-11-30.
Object id is 4506, canonical name is ValuationDomain.
Accessed 3226 times total.

Classification:

AMS MSC:	16U10 (Associative rings and algebras :: Conditions on elements :: Integral domains)
	13G05 (Commutative rings and algebras :: Integral domains)
	13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)

Pending Errata and Addenda

None.[ View all 2 ]

Discussion

forum policy

No messages.

Interact