# Universally catenary ring

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