Universally catenary ring

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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 R is a universally catenary ring and I is an ideal in R. Then, the quotient ring R/I 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