# Dedekind domain

A is a commutative integral domain $R$ for which:

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 $K$ 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 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