Bezout domain - Revision history

Vipul: /* Relation with other properties */

2014-01-29T20:23:17Z

Relation with other properties

← Older revision		Revision as of 20:23, 29 January 2014
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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::Multi-stage Euclidean domain]]		\|-
			\| [[Weaker than::Euclidean domain]] \|\| admits a [[Euclidean norm]]\|\| \|\| \|\| {{intermediate notions short\|Bezout domain\|Euclidean domain}}
			\|-
			\| [[Weaker than::Principal ideal domain]] \|\| every [[ideal]] is a [[principal ideal]] \|\| \|\| \|\| {{intermediate notions short\|Bezout domain\|principal ideal domain}}
			\|-
			\| [[Weaker than::Multi-stage Euclidean domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Bezout domain\|multi-stage Euclidean domain}}
			\|}

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

	* [[Stronger than::gcd domain]]~~: {{proofofstrictimplicationat~~\|[[Bezout implies gcd]]\|[[gcd not implies Bezout]]}}		{\| class="sortable" border="1"
	* [[Stronger than::Bezout ring]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::gcd domain]] \|\| any two elements have a well-defined gcd \|\| [[Bezout implies gcd]] \|\| [[gcd not implies Bezout]] \|\| {{intermediate notions short\|gcd domain\|Bezout domain}}
			\|-
			\| [[Stronger than::Bezout ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Bezout ring\|Bezout domain}}
			\|}

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

	* [[Principal ideal domain]] is the conjunction with the property of being a [[Noetherian ring]].		* [[Principal ideal domain]] is the conjunction with the property of being a [[Noetherian ring]].

Vipul: /* Relation with other properties */

2009-01-17T17:19:15Z

Relation with other properties

← Older revision		Revision as of 17:19, 17 January 2009
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	* [[Stronger than::gcd domain]]: {{proofofstrictimplicationat\|[[Bezout implies gcd]]\|[[gcd not implies Bezout]]}}		* [[Stronger than::gcd domain]]: {{proofofstrictimplicationat\|[[Bezout implies gcd]]\|[[gcd not implies Bezout]]}}
			* [[Stronger than::Bezout ring]]

			===Conjunction with other properties===

			* [[Principal ideal domain]] is the conjunction with the property of being a [[Noetherian ring]].

Vipul: /* Relation with other properties */

2009-01-17T17:18:19Z

Relation with other properties

← Older revision		Revision as of 17:18, 17 January 2009
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	===Stronger properties===		===Stronger properties===

	* [[Principal ideal domain]]		* [[Weaker than::Euclidean domain]]
	* [[Multi-stage Euclidean domain]]		* [[Weaker than::Principal ideal domain]]
			* [[Weaker than::Multi-stage Euclidean domain]]

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

	* [[gcd domain]]: {{proofofstrictimplicationat\|[[Bezout implies gcd]]\|[[gcd not implies Bezout]]}}		* [[Stronger than::gcd domain]]: {{proofofstrictimplicationat\|[[Bezout implies gcd]]\|[[gcd not implies Bezout]]}}

Vipul: 4 revisions

2008-05-12T16:18:50Z

4 revisions

← Older revision	Revision as of 16:18, 12 May 2008
(No difference)

Vipul: /* Weaker properties */

2008-01-05T21:09:23Z

Weaker properties

← Older revision		Revision as of 21:09, 5 January 2008
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	===Weaker properties===		===Weaker properties===

	* [[gcd domain]]: {{~~proofat~~\|[[Bezout implies gcd]]}}		* [[gcd domain]]: {{proofofstrictimplicationat\|[[Bezout implies gcd]]\|[[gcd not implies Bezout]]}}

Vipul: `

2008-01-05T20:44:47Z

← Older revision		Revision as of 20:44, 5 January 2008
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	===Weaker properties===		===Weaker properties===

	* [[gcd domain]]		* [[gcd domain]]: {{proofat\|[[Bezout implies gcd]]}}

Vipul at 23:15, 16 December 2007

2007-12-16T23:15:29Z

← Older revision		Revision as of 23:15, 16 December 2007
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	An [[integral domain]] is termed a '''Bezout domain''' if every [[finitely generated ideal]] in it is [[principal ideal\|principal]].		An [[integral domain]] is termed a '''Bezout domain''' if every [[finitely generated ideal]] in it is [[principal ideal\|principal]].

			==Relation with other properties==

			===Stronger properties===

			* [[Principal ideal domain]]
			* [[Multi-stage Euclidean domain]]

			===Weaker properties===

			* [[gcd domain]]

Vipul at 22:37, 16 December 2007

2007-12-16T22:37:05Z

New page

{{integral domain property}}

==Definition==

===Symbol-free definition===

An [[integral domain]] is termed a '''Bezout domain''' if every [[finitely generated ideal]] in it is [[principal ideal|principal]].