Universally catenary ring - Revision history

Vipul at 16:10, 18 July 2013

2013-07-18T16:10:45Z

@@ Line 7: / Line 7: @@
 Since catenary rings are, by definition, Noetherian, so are universally catenary rings.
 ==Relation with other properties==
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 | [[Stronger than::Noetherian ring]] || || || || {{intermediate notions short|Noetherian ring|universally catenary ring}}
 |}

Vipul: /* Weaker properties */

2013-07-18T16:09:06Z

Weaker properties

← Older revision		Revision as of 16:09, 18 July 2013
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	\|-		\|-
	\| [[Stronger than::Noetherian ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|universally catenary ring}}		\| [[Stronger than::Noetherian ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|universally catenary ring}}
			\|}

	==Metaproperties==		==Metaproperties==

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

Vipul at 16:08, 18 July 2013

2013-07-18T16:08:58Z

← Older revision		Revision as of 16:08, 18 July 2013
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	===Stronger properties===		===Stronger properties===

	* [[Cohen-Macaulay ring]]		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::Cohen-Macaulay ring]] \|\| for every [[ideal]], the [[depth]] equals the [[codimension]]. \|\| \|\| \|\| {{intermediate notions short\|universally catenary ring\|Cohen-Macaulay ring}}
			\|}

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

	* [[~~Catenary~~ ring]]		{\| class="sortable" border="1"
	* [[Noetherian ring]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::catenary ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|catenary ring\|universally catenary ring}}
			\|-
			\| [[Stronger than::Noetherian ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|universally catenary ring}}

	==Metaproperties==		==Metaproperties==

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

Vipul: 4 revisions

2008-05-12T16:35:07Z

4 revisions

← Older revision	Revision as of 16:35, 12 May 2008
(No difference)

Vipul: /* Definition */

2008-03-09T21:54:50Z

Definition

← Older revision		Revision as of 21:54, 9 March 2008
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	==Definition==		==Definition==

	A commutative unital ring is termed '''universally catenary''' if it ~~satisfies the following equivalent conditions:~~		A [[commutative unital ring]] is termed '''universally catenary''' if every [[finitely generated algebra]] over it is a [[catenary ring]].

	* Every [[finitely generated algebra]] over it is a [[catenary ~~ring]]~~		Since catenary rings are, by definition, Noetherian, so are universally catenary rings.
	* Every polynomial ring in finitely many variables, ~~over it~~, ~~is a~~ catenary ~~ring~~

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

Vipul at 00:42, 8 January 2008

2008-01-08T00:42:18Z

← Older revision		Revision as of 00:42, 8 January 2008
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	* [[Catenary ring]]		* [[Catenary ring]]
	* [[Noetherian ring]]		* [[Noetherian ring]]

			==Metaproperties==

			{{Q-closed curing property}}

Vipul at 00:35, 8 January 2008

2008-01-08T00:35:17Z

← Older revision		Revision as of 00:35, 8 January 2008
Line 3:		Line 3:
	==Definition==		==Definition==

	A commutative unital ring is termed '''universally catenary''' if ~~every~~ [[finitely generated algebra]] over it is a [[catenary ring]].		A commutative unital ring is termed '''universally catenary''' if it satisfies the following equivalent conditions:

			* Every [[finitely generated algebra]] over it is a [[catenary ring]]
			* Every polynomial ring in finitely many variables, over it, is a catenary ring

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

Vipul at 00:07, 8 January 2008

2008-01-08T00:07:53Z

New page

{{curing property}}

==Definition==

A commutative unital ring is termed '''universally catenary''' if every [[finitely generated algebra]] over it is a [[catenary ring]].

==Relation with other properties==

===Stronger properties===

* [[Cohen-Macaulay ring]]

===Weaker properties===

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