Dedekind domain
A Dedekind domain^{} is a commutative^{} integral domain^{} $R$ for which:

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Every ideal in $R$ is finitely generated^{}.

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Every nonzero prime ideal^{} is a maximal ideal^{}.

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The domain $R$ 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 1dimensional.
Every Dedekind domain is a noetherian ring^{}.
If $K$ is a number field, then ${\mathcal{O}}_{K}$, the ring of algebraic integers of $K$, is a Dedekind domain.
Title  Dedekind domain 
Canonical name  DedekindDomain 
Date of creation  20130322 12:36:06 
Last modified on  20130322 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 