Dedekind domain

A Dedekind domainMathworldPlanetmath is a commutativePlanetmathPlanetmathPlanetmathPlanetmath integral domainMathworldPlanetmath 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 setMathworldPlanetmath is a Dedekind domain if and only if the set is normal, irreduciblePlanetmathPlanetmathPlanetmathPlanetmath, and 1-dimensional.

Every Dedekind domain is a noetherian ringMathworldPlanetmath.

If K is a number field, then 𝒪K, 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