---
tag: B2CYR292J6
kind: concept
name: Catenary ring
msc: [13C15]
specializes: [0310REJJY8]
sources:
  - url: https://en.wikipedia.org/wiki/Catenary_ring
same_as:
  wikidata: Q5051871
  wikipedia: Catenary ring
---

## Summary

A catenary ring is a [[commutative ring|0310REJJY8]] in which, between any two nested [[prime ideals|00MHHZ9SNE]], all maximal chains of primes have the same finite length.
In such a ring the distance between two primes is well defined.
Most rings met in practice are catenary, such as finitely generated algebras over a field and Cohen–Macaulay rings.
Nagata showed that some Noetherian local domains are not catenary.

## Explanations

### L1 human

Think of the prime ideals as floors of a building, ordered by inclusion.
A chain of primes is a staircase, and a maximal chain is one with no floor skipped.
The ring is catenary when every staircase between two given floors has the same number of steps.
For example, in $\Z$ the only chains from $(0)$ to a prime $(p)$ are $(0) \subsetneq (p)$, all of length $1$.
The [[definition|K6PPGRT5E1]] makes this precise.