Dedekind domain
A Dedekind domain is a commutative integral domain for which:
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Every ideal in is finitely generated.
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Every nonzero prime ideal is a maximal ideal.
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The domain is integrally closed in its field of fractions.
It is worth noting that the second clause above implies that the maximal length of a strictly increasing chain of prime ideals is 1, so the Krull dimension of any Dedekind domain is at most 1. In particular, the affine ring of an algebraic set is a Dedekind domain if and only if the set is normal, irreducible, and 1-dimensional.
Every Dedekind domain is a noetherian ring.
If is a number field, then , the ring of algebraic integers of , is a Dedekind domain.
Title | Dedekind domain |
Canonical name | DedekindDomain |
Date of creation | 2013-03-22 12:36:06 |
Last modified on | 2013-03-22 12:36:06 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 16 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 11R37 |
Classification | msc 11R04 |
Related topic | IntegralClosure |
Related topic | PruferDomain |
Related topic | MultiplicationRing |
Related topic | PrimeIdealFactorizationIsUnique |
Related topic | EquivalentCharacterizationsOfDedekindDomains |
Related topic | ProofThatADomainIsDedekindIfItsIdealsAreInvertible |
Related topic | ProofThatADomainIsDedekindIfItsIdealsAreProductsOfPrimes |
Related topic | ProofThatADomai |