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Localization preserves Noetherian rings

If a ring is Noetherian, then any localization at a multiplicative set is again Noetherian.

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 .

Let RR be a and let SS be a in RR. Then the localized ring

S−1RS^{-1}R

is Noetherian.

More generally, if MM is a Noetherian RR-module, then S−1MS^{-1}M is a Noetherian S−1RS^{-1}R-module.

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

Examples
  1. Localizing Z\mathbb Z. The ring Z\mathbb Z is Noetherian, so any localization is Noetherian. For instance, the local ring Z(p)\mathbb Z_{(p)} (obtained by ) is Noetherian.
  1. Localizing a polynomial ring. Let R=k[x,y]R=k[x,y], which is Noetherian. Localizing at S={1,x,x2,… }S=\{1,x,x^2,\dots\} gives Rx=k[x,y]xR_x=k[x,y]_x, which is still Noetherian. Geometrically, this corresponds to restricting from affine space to the principal open set where x≠0x\neq 0 in with its .
  1. Non-Noetherian rings can stay non-Noetherian after localization. Let R=k[x1,x2,… ]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: Rx1R_{x_1} still contains an infinite strictly ascending chain of ideals (x2)⊊(x2,x3)⊊⋯(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 , then analyze invariants (dimension, depth, primary decomposition, and so on) locally.