Catenary ring - Revision history

Vipul at 16:02, 18 July 2013

2013-07-18T16:02:53Z

← Older revision		Revision as of 16:02, 18 July 2013
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	===Stronger properties===		===Stronger properties===

	* [[~~Polynomial~~ ring over a field]]		{\| class="sortable" border="1"
	* [[~~Affine~~ ring over a field]]: {{~~proofat~~\|[[~~Affine~~ implies catenary]]}}		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[~~Principal~~ ideal domain]]		\|-
	* [[~~Universally~~ catenary ring]]		\| [[Weaker than::polynomial ring over a field]] \|\| of the form <math>K[x]</math> where <math>K</math> is a [[field]] \|\| \|\| \|\| {{intermediate notions short\|catenary ring\|polynomial ring over a field}}
	* [[Cohen-Macaulay ring]]		\|-
			\| [[Weaker than::multivariate polynomial ring over a field]] \|\| of the form <math>K[x_1,x_2,\dots,x_n]</matH> where <math>K</math> is a [[field]] \|\| \|\| \|\| {{intermediate notions short\|catenary ring\|multivariate polynomial ring over a field}}
			\|-
			\| [[Weaker than::affine ring over a field]] \|\| \|\| [[affine implies catenary]] \|\| \|\| {{intermediate notions short\|catenary ring\|affine ring over a field}}
			\|-
			\| [[Weaker than::principal ideal domain]] \|\| [[integral domain]] in which every [[ideal]] is a [principal ideal]] \|\| \|\| {{intermediate notions short\|catenary ring\|principal ideal domain}}
			\|-
			\| [[Weaker than::universally catenary ring]] \|\| every finitely generated algebra over it is a catenary ring.\|\| \|\| \|\| {{intermediate notions short\|catenary ring\|universally catenary ring}}
			\|-
			\| [[Weaker than::Cohen-Macaulay ring]] \|\| for every ideal, the depth equals the codimension \|\| \|\| \|\| {{intermediate notions short\|catenary ring\|Cohen-Macaulay ring}}
			\|}
	===Weaker properties===		===Weaker properties===

	* [[Noetherian ring]]		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::Noetherian ring]] \|\| every [[ideal]] in it is [[finitely generated ideal\|finitely generated]] \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|catenary ring}}
			\|}

	==Metaproperties==		==Metaproperties==

	{{Q-closed curing property}}		{{Q-closed curing property}}

Vipul: 12 revisions

2008-05-12T16:19:04Z

12 revisions

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

Vipul at 16:08, 10 March 2008

2008-03-10T16:08:37Z

← Older revision		Revision as of 16:08, 10 March 2008
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	==Metaproperties==		==Metaproperties==

			{{Q-closed curing property}}

Vipul at 21:53, 9 March 2008

2008-03-09T21:53:22Z

← Older revision		Revision as of 21:53, 9 March 2008
Line 25:		Line 25:

	==Metaproperties==		==Metaproperties==

	~~{{Q-closed curing property}}~~

Vipul at 20:49, 9 March 2008

2008-03-09T20:49:35Z

← Older revision		Revision as of 20:49, 9 March 2008
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	==Definition==		==Definition==

	A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring\|Noetherian]] and satisfies the following ~~condition~~:		A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring\|Noetherian]] and satisfies the following equivalent conditions:

	If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>.		* If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>
			* Given two prime ideals <math>P</math> and <math>Q</math> such that <math>P \subset Q</math>, the length of any saturated chain of primes between <math>P</math> and <math>Q</math> (i.e. a chain of primes in which no more primes can be inserted in between) is determined independent of the choice of chain

			Note that being catenary does ''not'' guarantee that any two fully saturated chains of primes (i.e. any two chains of primes which are saturated and cannot be extended in either direction) have the same length. The problem is that the starting and ending points of the chains may differ: there may be many different [[maximal ideal]]s and many different [[minimal prime ideal]]s.

	==Relation with other properties==		==Relation with other properties==

Vipul at 00:36, 8 January 2008

2008-01-08T00:36:55Z

← Older revision		Revision as of 00:36, 8 January 2008
Line 3:		Line 3:
	==Definition==		==Definition==

	A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring\|Noetherian]] satisfies the following condition:		A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring\|Noetherian]] and satisfies the following condition:

	If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>.		If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>.
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	* [[Noetherian ring]]		* [[Noetherian ring]]

			==Metaproperties==

			{{Q-closed curing property}}

Vipul at 00:33, 8 January 2008

2008-01-08T00:33:13Z

← Older revision		Revision as of 00:33, 8 January 2008
Line 3:		Line 3:
	==Definition==		==Definition==

	A [[commutative unital ring]] is said to be '''~~catenary~~''' if it is [[Noetherian ring\|Noetherian]] satisfies the following condition:		A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring\|Noetherian]] satisfies the following condition:

	If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>.		If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>.

Vipul: /* Stronger properties */

2008-01-08T00:07:32Z

Stronger properties

← Older revision		Revision as of 00:07, 8 January 2008
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	* [[Affine ring over a field]]: {{proofat\|[[Affine implies catenary]]}}		* [[Affine ring over a field]]: {{proofat\|[[Affine implies catenary]]}}
	* [[Principal ideal domain]]		* [[Principal ideal domain]]
			* [[Universally catenary ring]]
			* [[Cohen-Macaulay ring]]

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

	* [[Noetherian ring]]		* [[Noetherian ring]]

Vipul: /* Relation with other properties */

2008-01-08T00:00:48Z

Relation with other properties

← Older revision		Revision as of 00:00, 8 January 2008
Line 11:		Line 11:
	===Stronger properties===		===Stronger properties===

	* [[Affine ring]]: {{proofat\|[[Affine implies catenary]]}}		* [[Polynomial ring over a field]]
			* [[Affine ring over a field]]: {{proofat\|[[Affine implies catenary]]}}
			* [[Principal ideal domain]]
	===Weaker properties===		===Weaker properties===

	* [[Noetherian ring]]		* [[Noetherian ring]]

Vipul: /* Relation with other properties */

2008-01-07T23:58:51Z

Relation with other properties

← Older revision		Revision as of 23:58, 7 January 2008
Line 11:		Line 11:
	===Stronger properties===		===Stronger properties===

	* [[Affine ring]]		* [[Affine ring]]: {{proofat\|[[Affine implies catenary]]}}

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

	* [[Noetherian ring]]		* [[Noetherian ring]]