# Difference between revisions of "Dedekind domain"

From Commalg

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===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

− | + | |- | |

− | + | | [[Weaker than::Euclidean domain]] || [[integral domain]] that admits a [[Euclidean norm]] || || || {{intermediate notions short|Dedekind domain|Euclidean domain}} | |

+ | |- | ||

+ | | [[Weaker than::principal ideal domain]] || [[integral domain]] in which every [[ideal]] is a [[principal ideal]] || || [[Dedekind not implies PID]] || {{intermediate notions short|principal ideal domain|Dedekind domain}} | ||

+ | |- | ||

+ | | [[Weaker than::polynomial ring over a field]] || ring of the form <math>K[x]</math> where <math>K</math> is a field || || || {{intermediate notions short|Dedekind domain|polynomial ring over a field}} | ||

+ | |- | ||

+ | | [[Weaker than::ring of integers in a number field]] || it is the integral closure of <math>\mathbb{Z}</math> inside a [[number field]] (a finite degree field extension of the rationals) || || || {{intermediate notions short|Dedekind domain|ring of integers in a number field}} | ||

+ | |} | ||

===Weaker properties=== | ===Weaker properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

− | + | |- | |

− | + | | [[Stronger than::normal ring]] || [[integrally closed subring|integrally]] in its [[total quotient ring]] || || || {{intermediate notions short|normal ring|Dedekind domain}} | |

− | + | |- | |

− | + | | [[Stronger than::normal domain]] || [[integrally closed subring|integrally closed]] in its [[field of fractions]] || || || {{intermediate notions short|normal domain|Dedekind domain}} | |

− | + | |- | |

− | + | | [[Stronger than::Noetherian ring]] || every [[ideal]] is [[finitely generated ideal|finitely generated]] || || || {{intermediate notions short|Noetherian ring|Dedekind domain}} | |

+ | |- | ||

+ | | [[Stronger than::one-dimensional ring]] || the [[Krull dimension]] is at most one, i.e., we cannot have a strictly ascending chain <math>P_0 \subset P_1 \subset P_2</math> of prime ideals || || || {{intermediate notions short|one-dimensional ring|Dedekind domain}} | ||

+ | |- | ||

+ | | [[Stronger than::Noetherian domain]] || [[integral domain]] and a [[Noetherian ring]] || || || {{intermediate notions short|Noetherian domain|Dedekind domain}} | ||

+ | |- | ||

+ | | [[Stronger than::one-dimensional domain]] || [[integral domain]] and a [[one-dimensional ring]]. Explicitly the prime ideals are precisely the zero ideal and nonzero maximal ideals. || || || {{intermediate notions short|one-dimensional domain|Dedekind domain}} | ||

+ | |- | ||

+ | | [[Stronger than::one-dimensional Noetherian domain]] || [[integral domain]] that is a [[Noetherian ring]] and a [[one-dimensional ring]]. || || || {{intermediate notions short|one-dimensional Noetherian domain|Dedekind domain}} | ||

+ | |- | ||

+ | | [[Stronger than::Noetherian normal domain]] || [[integral domain]] that is a [[Noetherian ring]] and a [[normal ring]]. || || || {{intermediate notions short|Noetherian normal domain|Dedekind domain}} | ||

+ | |} | ||

===Conjunction with other properties=== | ===Conjunction with other properties=== |

## Latest revision as of 03:51, 18 July 2013

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 .