---
tag: KTQC33ZJTN
kind: definition
name: Regular local ring
msc: [13H05]
defines: 63XXTF6B5M
uses: [MZ02BKPHJA, 0670T59HGE, MVGBS3J1P7, DP3KSCRKFP]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_local_ring
---

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with residue field $k = R/\mathfrak m$, and let $d = \dim R$ be its [[Krull dimension|MVGBS3J1P7]].
The ring $R$ is *regular* if $\mathfrak m$ can be generated by $d$ elements.
Equivalently, $R$ is regular if its [[embedding dimension|DP3KSCRKFP]] $\dim_k \mathfrak m / \mathfrak m^2$ equals $d$.

## Remarks

A set of $d$ generators of $\mathfrak m$ in a regular local ring is called a *regular system of parameters*.

Example: a field is regular, with $d = 0$ and $\mathfrak m = 0$.
A Noetherian local domain of dimension $1$ is regular exactly when $\mathfrak m$ is principal, that is, when it is a [[discrete valuation ring|MGM3CNJW9V]], such as $\Z_{(p)}$.
For a field $k$, the rings $k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}$ and $k[\![x_1, \ldots, x_n]\!]$ have dimension $n$ and maximal ideal generated by the $n$ elements $x_1, \ldots, x_n$, so they are regular.

Non-example: $R = k[x]/(x^2)$ has dimension $0$, but its maximal ideal $(x) \ne 0$ needs one generator, so it is not regular.
The cusp $k[x, y]_{(x, y)}/(y^2 - x^3)$ has dimension $1$ and embedding dimension $2$, so it is not regular.