Dedekind domain

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

• 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.