Dedekind domain

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

• Every ideal in $R$ is finitely generated.

• Every nonzero prime ideal is a maximal ideal.

• 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 1-dimensional.

Every Dedekind domain is a noetherian ring.

If $K$ is a number field, then ${𝒪}_K$, 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