Difference between revisions of "Universally catenary ring"

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

Revision as of 16:08, 18 July 2013

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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

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

Since catenary rings are, by definition, Noetherian, so are universally catenary rings.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Cohen-Macaulay ring for every ideal, the depth equals the codimension. click here

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
catenary ring click here
Noetherian ring click here

Metaproperties

Closure under taking quotient rings

This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this property


View other quotient-closed properties of commutative unital rings