Localization preserves Noetherian rings

Localization is a “local” operation, but it does not destroy finiteness properties: in particular, Noetherianity survives. This is indispensable when passing from global to local statements, for example by localizing at a prime .

Theorem

Let $R$ be a Noetherian ring and let $S$ be a multiplicative set in $R$. Then the localized ring

is Noetherian.

More generally, if $M$ is a Noetherian $R$-module, then $S^{-1}M$ is a Noetherian $S^{-1}R$-module.

This result is often paired with exactness of localization , since many Noetherian arguments proceed by short exact sequences and localization.

Examples

1. Localizing $\mathbb Z$.
   The ring $\mathbb Z$ is Noetherian, so any localization is Noetherian. For instance, the local ring $\mathbb Z_{(p)}$ (obtained by localizing at the prime $(p)$ ) is Noetherian.

2. Localizing a polynomial ring.
   Let $R=k[x,y]$, which is Noetherian. Localizing at $S=\{1,x,x^2,\dots\}$ gives $R_x=k[x,y]_x$, which is still Noetherian. Geometrically, this corresponds to restricting from affine space to the principal open set where $x\neq 0$ in Spec with its Zariski topology .

3. Non-Noetherian rings can stay non-Noetherian after localization.
   Let $R=k[x_1,x_2,\dots]$, which is not Noetherian. Localizing at many natural multiplicative sets (for example, inverting one variable) typically does not repair this: $R_{x_1}$ still contains an infinite strictly ascending chain of ideals $(x_2)\subsetneq(x_2,x_3)\subsetneq\cdots$. This illustrates that the theorem is one-way: localization preserves Noetherianity, but does not create it.

When applying the theorem in practice, a common pattern is: start with a Noetherian ring, localize at a prime to get a Noetherian local ring , then analyze invariants (dimension, depth, primary decomposition, and so on) locally.