Universally catenary ring: Difference between revisions
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Since catenary rings are, by definition, Noetherian, so are universally catenary rings. | Since catenary rings are, by definition, Noetherian, so are universally catenary rings. | ||
==Metaproperties== | |||
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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | |||
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| [[satisfies metaproperty::quotient-closed property of commutative unital rings]] || Yes || [[universally catenary is quotient-closed]] || Suppose <math>R</math> is a universally catenary ring and <math>I</math> is an [[ideal]] in <math>R</math>. Then, the [[quotient ring]] <math>R/I</math> is also a universally catenary ring. | |||
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==Relation with other properties== | ==Relation with other properties== | ||
===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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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=== | ||
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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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{{ | | [[Stronger than::Noetherian ring]] || || || || {{intermediate notions short|Noetherian ring|universally catenary ring}} | ||
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Latest revision as of 16:10, 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.
Metaproperties
| Metaproperty name | Satisfied? | Proof | Statement with symbols |
|---|---|---|---|
| quotient-closed property of commutative unital rings | Yes | universally catenary is quotient-closed | Suppose is a universally catenary ring and is an ideal in . Then, the quotient ring is also a universally catenary ring. |
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 |