Dedekind domain - Revision history

Vipul: /* Relation with other properties */

2013-07-18T03:51:04Z

Relation with other properties

← Older revision		Revision as of 03:51, 18 July 2013
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::Euclidean domain]]		{\| class="sortable" border="1"
	* [[Weaker than::~~Principal~~ ideal domain]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::~~Polynomial~~ ring over a field]]		\|-
	* [[Weaker than::~~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===		===Weaker properties===

	* [[Stronger than::~~Normal~~ ring]]		{\| class="sortable" border="1"
	* [[Stronger than::~~Noetherian ring~~]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Stronger than::~~One-dimensional~~ ring]]		\|-
	* [[Stronger than::~~Normal domain~~]]		\| [[Stronger than::normal ring]] \|\| [[integrally closed subring\|integrally]] in its [[total quotient ring]] \|\| \|\| \|\| {{intermediate notions short\|normal ring\|Dedekind domain}}
	* [[Stronger than::Noetherian domain]]		\|-
	* [[Stronger than::~~One~~-dimensional domain]]		\| [[Stronger than::normal domain]] \|\| [[integrally closed subring\|integrally closed]] in its [[field of fractions]] \|\| \|\| \|\| {{intermediate notions short\|normal domain\|Dedekind domain}}
	* [[Stronger than::~~One~~-dimensional Noetherian domain]]		\|-
	* [[Stronger than::Noetherian normal 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===

Vipul at 17:17, 17 January 2009

2009-01-17T17:17:24Z

← Older revision		Revision as of 17:17, 17 January 2009
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	===Stronger properties===		===Stronger properties===

	* [[Euclidean domain]]		* [[Weaker than::Euclidean domain]]
	* [[Principal ideal domain]]		* [[Weaker than::Principal ideal domain]]
	* [[Polynomial ring over a field]]		* [[Weaker than::Polynomial ring over a field]]
	* [[Ring of integers in a number field]]		* [[Weaker than::Ring of integers in a number field]]

	===Weaker properties===		===Weaker properties===

	* [[Normal domain]]		* [[Stronger than::Normal ring]]
	* [[Noetherian domain]]		* [[Stronger than::Noetherian ring]]
	* [[One-dimensional domain]]		* [[Stronger than::One-dimensional ring]]
			* [[Stronger than::Normal domain]]
			* [[Stronger than::Noetherian domain]]
			* [[Stronger than::One-dimensional domain]]
			* [[Stronger than::One-dimensional Noetherian domain]]
			* [[Stronger than::Noetherian normal domain]]

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

Vipul: 3 revisions

2008-05-12T16:19:33Z

3 revisions

← Older revision	Revision as of 16:19, 12 May 2008
(No difference)

Vipul at 21:05, 20 January 2008

2008-01-20T21:05:44Z

← Older revision		Revision as of 21:05, 20 January 2008
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	==Metaproperties==		==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>.

Vipul at 20:54, 20 January 2008

2008-01-20T20:54:13Z

← Older revision		Revision as of 20:54, 20 January 2008
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	* It is a [[Noetherian ring\|Noetherian]] [[normal domain]] of [[Krull dimension]] 1		* It is a [[Noetherian ring\|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 ideals\|product]] of [[prime ideal]]s		* Every nonzero ideal is invertible in the field of fractions and can be expressed uniquely as a [[product of ideals\|product]] of [[prime ideal]]s

			==Relation with other properties==

			===Stronger properties===

			* [[Euclidean domain]]
			* [[Principal ideal domain]]
			* [[Polynomial ring over a field]]
			* [[Ring of integers in a number field]]

			===Weaker properties===

			* [[Normal domain]]
			* [[Noetherian domain]]
			* [[One-dimensional domain]]

			===Conjunction with other properties===

			Any [[unique factorization domain]] which is also a Dedekind domain, is also a [[principal ideal domain]].

			==Metaproperties==

Vipul at 10:05, 8 August 2007

2007-08-08T10:05:16Z

New page

{{integral domain property}}

==Definition==

===Symbol-free definition===

An [[integral domain]] is termed a '''Dedekind domain''' if it satisfies the following equivalent conditions:

* It is a [[Noetherian ring|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 ideals|product]] of [[prime ideal]]s