valuation domain
An integral domain is a valuation domain if for all , either or . Equivalently, an integral domain is a valuation domain if for any in the field of fractions of , .
Some properties:
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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.
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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.
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Valuation domains are integrally closed.
Title | valuation domain |
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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 |