---
tag: 68E0F8N94E
kind: concept
name: Localization of a module
msc: [13B30, 13C99]
sources:
  - url: https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
---

## Summary

Let $M$ be a [[module|0NZ8JA25WK]] over a [[commutative ring|0310REJJY8]] $R$, and let $S$ be a [[multiplicative set|AN9SKEZ3VY]].
The localization of $M$ at $S$ is the module $S^{-1}M$ of fractions $m/s$ with $m \in M$ and $s \in S$.
It is a module over the [[localization|4C1AE84N5N]] $S^{-1}R$.
Localization of modules preserves [[exact sequences|JB7ZBFBA9C]], so it is a basic tool to reduce statements about modules to the local case.

## Explanations

### L1 human

A module is like a vector space whose scalars come from a ring.
Localizing the module means allowing the same new denominators as in the ring.
Elements killed by some allowed denominator become zero.
For example, over $\Z$ the module $\Z/2\Z$ disappears once you may divide by $2$, because $1 = 2 \cdot (1/2)$ and $2 \cdot 1 = 0$ there.
The [[definition|S357810Z25]] gives the construction.