---
tag: C618HAK3X6
kind: statement
role: proposition
name: Localization of a Noetherian ring is Noetherian
msc: ["13E05", 13B30]
uses: [0670T59HGE, 4C1AE84N5N, AN9SKEZ3VY, 5AFH94P4S3]
about: [0670T59HGE, 4C1AE84N5N]
sources:
  - url: https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
---

## Statement

Let $R$ be a [[Noetherian|0670T59HGE]] commutative ring and let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]].
Then the [[localization|4C1AE84N5N]] $S^{-1}R$ is a Noetherian ring.

## Remarks

Let $\lambda \colon R \to S^{-1}R$ be the canonical map.
Each ideal $J$ of $S^{-1}R$ is the [[extension|5AFH94P4S3]] of its contraction: $J = S^{-1}\lambda^{-1}(J)$.
Indeed, if $a/s \in J$, then $a/1 = s \cdot (a/s) \in J$, so $a \in \lambda^{-1}(J)$.
Hence a strictly ascending chain of ideals in $S^{-1}R$ gives a strictly ascending chain of their contractions in $R$, which cannot be infinite.
The same argument shows that $S^{-1}M$ is a Noetherian $S^{-1}R$-module when $M$ is a Noetherian $R$-module.

Example: $\Z_{(p)}$, $\Z[1/n]$, $\Q$ and the local rings $k[x_1, \ldots, x_n]_{\mathfrak p}$ of a polynomial ring over a field $k$ are Noetherian.

The converse fails: Noetherianity is not a [[local property|3S58E70Z7Z]].
An infinite product $R = \prod_{i \in \N} k$ of copies of a field $k$ is not Noetherian, but each localization $R_{\mathfrak p}$ at a prime ideal is a field.

## Explanations

### L1 human

A ring is Noetherian when it has no infinite strictly increasing chain of ideals.
Allowing new denominators never creates such a chain, because every ideal of the new ring comes from an ideal of the old ring.
So the good finiteness property of the old ring passes to the new one.

### L2 human

This fact is used all the time with $S = R \setminus \mathfrak p$: the local rings $R_{\mathfrak p}$ of a Noetherian ring are Noetherian local rings.
Thus the theory of Noetherian local rings, such as [[Krull dimension|MVGBS3J1P7]], the [[embedding dimension|DP3KSCRKFP]] and [[regular local rings|63XXTF6B5M]], applies to every prime of a Noetherian ring.
Together with [[Hilbert's basis theorem|0YKA9J3XEE]] and passage to quotients, it shows that the coordinate rings of affine varieties and all their localizations are Noetherian.