Universally catenary ring
From Commalg
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
Contents
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
MetapropertiesClosure under taking quotient ringsThis 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 |