Dedekind domain
From Commalg
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
Contents
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
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Euclidean domain | integral domain that admits a Euclidean norm | click here | ||
principal ideal domain | integral domain in which every ideal is a principal ideal | Dedekind not implies PID | click here | |
polynomial ring over a field | ring of the form ![]() ![]() |
click here | ||
ring of integers in a number field | it is the integral closure of ![]() |
click here |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
normal ring | integrally in its total quotient ring | click here | ||
normal domain | integrally closed in its field of fractions | click here | ||
Noetherian ring | every ideal is finitely generated | click here | ||
one-dimensional ring | the Krull dimension is at most one, i.e., we cannot have a strictly ascending chain ![]() |
click here | ||
Noetherian domain | integral domain and a Noetherian ring | click here | ||
one-dimensional domain | integral domain and a one-dimensional ring. Explicitly the prime ideals are precisely the zero ideal and nonzero maximal ideals. | click here | ||
one-dimensional Noetherian domain | integral domain that is a Noetherian ring and a one-dimensional ring. | click here | ||
Noetherian normal domain | integral domain that is a Noetherian ring and a normal ring. | click here |
Conjunction with other properties
Any unique factorization domain which is also a Dedekind domain, is also a principal ideal domain.
Metaproperties
Module theory
Any finitely generated module over a Dedekind domain
can be expressed as a direct sum as follows:
where is an ascending chain of ideals, which could reach
.