# 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 a **catenary ring** or **chain ring** or is said to satisfy the **Saturated Chain Condition** if it is Noetherian and satisfies the following equivalent conditions:

- If is a strictly ascending chain of prime ideals, and is a prime ideal between and , then there is either a prime ideal between and or a prime ideal between and
- Given two prime ideals and such that , the length of any saturated chain of primes between and (i.e. a chain of primes in which no more primes can be inserted in between) is determined independent of the choice of chain

Note that being catenary does *not* guarantee that any two fully saturated chains of primes (i.e. any two chains of primes which are saturated and cannot be extended in either direction) have the same length. The problem is that the starting and ending points of the chains may differ: there may be many different maximal ideals and many different minimal prime ideals.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

polynomial ring over a field | of the form where is a field | click here | ||

multivariate polynomial ring over a field | of the form where is a field | click here | ||

affine ring over a field | affine implies catenary | click here | ||

principal ideal domain | integral domain in which every ideal is a [principal ideal]] | click here | ||

universally catenary ring | every finitely generated algebra over it is a catenary ring. | click here | ||

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 |
---|---|---|---|---|

Noetherian ring | every ideal in it is finitely generated | 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 propertyView other quotient-closed properties of commutative unital rings