# 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 ringsVIEW 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 where is a field | click here | ||

ring of integers in a number field | it is the integral closure of inside a number field (a finite degree field extension of the rationals) | 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 of prime ideals | 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 .