Catenary ring

In commutative algebra, a ring chain ring or a ring katenärer is called, if not refinable Primidealketten two nested prime ideals always have the same length. Catenary rings have simple dimensional theoretical properties.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definition

Is a ring, a is a sequence of prime ideals Primidealkette ():

The length of this Primidealkette. Such Primidealkette is no longer called refinable chain, if there is no prime ideal, so

A Primidealkette is.

If a ring, it is called katenär or even a chain ring if and only if for all prime ideals that all non- refinable Primidealketten that start with and end with, have the same length.

Properties

• Is a Noetherian ring katenär, then each residue class ring and each localization.
• Katenär is a local property: a Noetherian ring if and only katenär when the ring is katenär for each maximal ideal.
• If noetherian katenär and zero divisors, and also all maximal ideals of the same height (for example, see below), then also each residue class ring for a prime ideal of this property. Then for each prime ideal:

Examples

• A body, the ring is katenär.
• Every Cohen- Macaulay ring, in particular, any regular ring is katenär.