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 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 |