# 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 ringsVIEW 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.

## 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 |