Difference between revisions of "Dedekind domain"

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(→‎Relation with other properties)
 
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===Stronger properties===
 
===Stronger properties===
  
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* [[Weaker than::Euclidean domain]]
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{| class="sortable" border="1"
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* [[Weaker than::Principal ideal domain]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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* [[Weaker than::Polynomial ring over a field]]
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|-
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* [[Weaker than::Ring of integers in a number field]]
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| [[Weaker than::Euclidean domain]] || [[integral domain]] that admits a [[Euclidean norm]] || || || {{intermediate notions short|Dedekind domain|Euclidean domain}}
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|-
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| [[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}}
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|-
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| [[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}}
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|-
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| [[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}}
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|}
  
 
===Weaker properties===
 
===Weaker properties===
  
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* [[Stronger than::Normal ring]]
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{| class="sortable" border="1"
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* [[Stronger than::Noetherian ring]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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* [[Stronger than::One-dimensional ring]]
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|-
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* [[Stronger than::Normal domain]]
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| [[Stronger than::normal ring]] || [[integrally closed subring|integrally]] in its [[total quotient ring]] || || || {{intermediate notions short|normal ring|Dedekind domain}}
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* [[Stronger than::Noetherian domain]]
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|-
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* [[Stronger than::One-dimensional domain]]
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| [[Stronger than::normal domain]] || [[integrally closed subring|integrally closed]] in its [[field of fractions]] || || || {{intermediate notions short|normal domain|Dedekind domain}}
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* [[Stronger than::One-dimensional Noetherian domain]]
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|-
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* [[Stronger than::Noetherian normal domain]]
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| [[Stronger than::Noetherian ring]] || every [[ideal]] is [[finitely generated ideal|finitely generated]] || || || {{intermediate notions short|Noetherian ring|Dedekind domain}}
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|-
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| [[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}}
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|-
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| [[Stronger than::Noetherian domain]] || [[integral domain]] and a [[Noetherian ring]] || || || {{intermediate notions short|Noetherian domain|Dedekind domain}}
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|-
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| [[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}}
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|-
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| [[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}}
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|-
 +
| [[Stronger than::Noetherian normal domain]] || [[integral domain]] that is a [[Noetherian ring]] and a [[normal ring]]. || || || {{intermediate notions short|Noetherian normal domain|Dedekind domain}}
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|}
  
 
===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 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:

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 K[x] where K is a field click here
ring of integers in a number field it is the integral closure of \mathbb{Z} 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 P_0 \subset P_1 \subset P_2 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 M over a Dedekind domain R can be expressed as a direct sum as follows:

M \cong R/I_1 \oplus R/I_2 \oplus \ldots \oplus R/I_n

where I_1 \subset I_2 \subset \ldots \subset I_n is an ascending chain of ideals, which could reach R.

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