Dedekind domain: Difference between revisions

Revision as of 20:54, 20 January 2008

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Symbol-free definition

An integral domain is termed a Dedekind domain if it satisfies the following equivalent conditions:

It is a Noetherian normal domain of Krull dimension 1
Every nonzero ideal is invertible in the field of fractions and can be expressed uniquely as a product of prime ideals

Relation with other properties

Stronger properties

Weaker properties

Conjunction with other properties

Any unique factorization domain which is also a Dedekind domain, is also a principal ideal domain.

Metaproperties

@@ Line 9: / Line 9: @@
 * It is a [[Noetherian ring|Noetherian]] [[normal domain]] of [[Krull dimension]] 1
 * Every nonzero ideal is invertible in the field of fractions and can be expressed uniquely as a [[product of ideals|product]] of [[prime ideal]]s
+==Relation with other properties==
+===Stronger properties===
+* [[Euclidean domain]]
+* [[Principal ideal domain]]
+* [[Polynomial ring over a field]]
+* [[Ring of integers in a number field]]
+===Weaker properties===
+* [[Normal domain]]
+* [[Noetherian domain]]
+* [[One-dimensional domain]]
+===Conjunction with other properties===
+Any [[unique factorization domain]] which is also a Dedekind domain, is also a [[principal ideal domain]].
+==Metaproperties==