Localization of a Noetherian ring is Noetherian

Corollary (Noetherian rings localize to Noetherian rings).
Let $R$ be a Noetherian ring and let $S\subseteq R$ be a multiplicative set . Then the localized ring

is Noetherian.

In particular, for any prime ideal $\mathfrak p\subset R$, the localization

(from localization at a prime ) is a Noetherian local ring .

This is an immediate consequence of localization preserves Noetherianity .

Related knowls.

• Localization of a ring
• Localization at a prime
• Local ring
• Noetherian ring

Examples

1. Localizing $\mathbb{Z}$ at a prime.
   Take $R=\mathbb{Z}$ and $S=\mathbb{Z}\setminus(p)$. Then $S^{-1}\mathbb{Z}=\mathbb{Z}_{(p)}$ is Noetherian (in fact a PID).

2. Inverting an element in a polynomial ring.
   Let $R=k[x,y]$ (Noetherian) and $S=\{1,x,x^2,\dots\}$. Then

   $$S^{-1}R \cong k[x,y]_x \cong k[x,y,1/x]$$

   is Noetherian.

3. Localizing at a prime ideal.
   In $R=k[x,y]$, let $\mathfrak p=(x)$. Then $R_{\mathfrak p}$ is Noetherian, and elements outside $(x)$ become units in $R_{\mathfrak p}$.