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 said to be catenary if it is Noetherian satisfies the following condition:

If $P<P_1<P_2<Q$ is a strictly ascending chain of prime ideals, and $P^{\prime }$ is a prime ideal between $P$ and $Q$, then there is either a prime ideal between $P$ and $P^{\prime }$ or a prime ideal between $P^{\prime }$ and $Q$.

Relation with other properties

Stronger properties

• Polynomial ring over a field
• Affine ring over a field: For full proof, refer: Affine implies catenary
• Principal ideal domain
• Universally catenary ring
• Cohen-Macaulay ring

Weaker properties

• Noetherian ring