Universally catenary ring: Difference between revisions
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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}} | ||
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==Metaproperties== | ==Metaproperties== | ||
{{Q-closed curing property}} | {{Q-closed curing property}} | ||
Revision as of 16:09, 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