---
tag: 63XXTF6B5M
kind: concept
name: Regular local ring
msc: [13H05]
specializes: [MZ02BKPHJA]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_local_ring
same_as:
  wikidata: Q2138875
  wikipedia: Regular local ring
---

## Summary

A regular local ring is a Noetherian [[local ring|MZ02BKPHJA]] whose maximal ideal can be generated by as many elements as its [[Krull dimension|MVGBS3J1P7]].
Equivalently, its [[embedding dimension|DP3KSCRKFP]] equals its dimension.
Regular local rings are the algebraic form of smooth points of varieties.
They are [[integral domains|V3ZBMRKC08]] and even [[unique factorization domains|1HD3EK5392]], and by [[Serre's theorem|93RQRG9R1P]] they are the Noetherian local rings of finite global dimension.

## Explanations

### L1 human

At a smooth point of a surface, two coordinates describe everything nearby, and the surface has dimension two.
At a singular point, like the tip of a cone, you need more coordinates than the dimension.
A regular local ring is the local ring of a point where the number of needed coordinates equals the dimension.
Examples are fields, the ring $\Z_{(p)}$, and power series rings $k[\![x_1, \ldots, x_n]\!]$.
The [[definition|KTQC33ZJTN]] gives the condition.