---
tag: YF7YFQ58QJ
kind: concept
name: Excellent ring
msc: [13F40, "13E05"]
specializes: [0670T59HGE]
sources:
  - url: https://en.wikipedia.org/wiki/Excellent_ring
same_as:
  wikidata: Q5419439
  wikipedia: Excellent ring
---

## Summary

An excellent ring is a [[Noetherian ring|0670T59HGE]] that satisfies three extra conditions on chains of primes, on completions and on regular loci.
The class was introduced by Grothendieck to collect the Noetherian rings that behave as well as the rings met in geometry and number theory.
Fields, the integers, complete Noetherian local rings and all finitely generated algebras over them are excellent.
For excellent rings, properties such as being reduced or normal pass from a local ring to its completion.
Also, the integral closure of a reduced excellent ring is a finite module over it.

## Explanations

### L1 human

Noetherian rings can be strange: some have maximal chains of primes of different lengths between the same two primes, and some local domains have a completion with nilpotent elements.
Excellent rings exclude these pathologies.
Almost every ring that appears in practice, such as a polynomial ring over a field or a quotient of it, is excellent.
So the word "excellent" marks the rings for which the usual geometric intuition works.