Dedekind domain: Difference between revisions

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:

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 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	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 $R$ .

@@ Line 14: / Line 14: @@
 ===Stronger properties===
-* [[Euclidean domain]]
+{| class="sortable" border="1"
-* [[Principal ideal domain]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Polynomial ring over a field]]
+|-
-* [[Ring of integers in a number field]]
+| [[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===
-* [[Normal domain]]
+{| class="sortable" border="1"
-* [[Noetherian domain]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[One-dimensional domain]]
+|-
+| [[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===
@@ Line 30: / Line 53: @@
 ==Metaproperties==
+==Module theory==
+Any [[finitely generated module]] <math>M</math> over a Dedekind domain <math>R</math> can be expressed as a direct sum as follows:
+<math>M \cong R/I_1 \oplus R/I_2 \oplus \ldots \oplus R/I_n</math>
+where <math>I_1 \subset I_2 \subset \ldots \subset I_n</math> is an ascending chain of ideals, which could reach <math>R</math>.