---
tag: 5WAFPTH09V
kind: concept
name: Universally catenary ring
msc: [13C15]
specializes: [B2CYR292J6]
sources:
  - url: https://en.wikipedia.org/wiki/Catenary_ring
---

## Summary

A universally catenary ring is a [[Noetherian ring|0670T59HGE]] whose finitely generated algebras are all [[catenary|B2CYR292J6]].
Being catenary is not preserved when one adjoins variables, so this stronger condition is the one that behaves well under the usual constructions.
Fields, $\Z$, Cohen–Macaulay rings and [[excellent rings|YF7YFQ58QJ]] are universally catenary.

## Explanations

### L1 human

A catenary ring is one where all maximal chains of prime ideals between two primes have the same length.
A universally catenary ring keeps this property after any finite number of polynomial variables are added and after any quotient is taken.
For example, every finitely generated algebra over a field, such as $k[x, y]/(xy)$, is catenary because the field $k$ is universally catenary.
The [[definition|PYABBA1QQE]] makes this precise.