---
tag: KR7SWBYHRQ
kind: concept
name: Normal ring
msc: [13B22]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_scheme
---

## Summary

A normal ring is a commutative ring whose [[localization|X6JD4AG973]] at every prime ideal is an [[integrally closed domain|H20P600HQS]].
It is the local form of integral closedness, and it allows rings that are not domains, such as products of integrally closed domains.
A Noetherian normal ring is a finite product of integrally closed domains.
Normal rings are the rings of functions on normal schemes, and normality is described by [[Serre's criterion|W27Z60Q1PF]].

## Explanations

### L1 human

Being integrally closed is a property of a domain, but a space can have several pieces, and its ring of functions is then not a domain.
Normality asks for integral closedness near each point, by looking at the ring of germs of functions there.
The ring $\Z \times \Z$ is normal, as it describes two disjoint copies of the same well-behaved space.
The ring $k[x, y]/(y^2 - x^3)$ of the cusp curve is not normal, because it fails at the cusp.