---
tag: S357810Z25
kind: definition
name: Localization of a module
msc: [13B30, 13C99]
defines: 68E0F8N94E
uses: [0NZ8JA25WK, AN9SKEZ3VY, 4C1AE84N5N, 0G8VEZ3WTK]
sources:
  - url: https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]], and let $M$ be an $R$-[[module|0NZ8JA25WK]].
On $M \times S$, let $(m, s) \sim (n, t)$ if there is $u \in S$ with $u(tm - sn) = 0$.
This is an [[equivalence relation|0G8VEZ3WTK]]; write $m/s$ for the class of $(m, s)$.
The *localization* of $M$ at $S$ is the set $S^{-1}M$ of these classes with the operations
$$\frac{m}{s} + \frac{n}{t} = \frac{tm + sn}{st}, \qquad \frac{a}{t} \cdot \frac{m}{s} = \frac{am}{ts}$$
for $a \in R$.
These operations are well defined and make $S^{-1}M$ a module over the [[localization|4C1AE84N5N]] $S^{-1}R$.
For a [[prime ideal|00MHHZ9SNE]] $\mathfrak p$ of $R$, one writes $M_{\mathfrak p}$ for $S^{-1}M$ with $S = R \setminus \mathfrak p$.

## Remarks

There is a natural isomorphism of $S^{-1}R$-modules $S^{-1}R \otimes_R M \cong S^{-1}M$, given by $(a/s) \otimes m \mapsto am/s$; see [[tensor product of modules|F0HQXQ0HCD]].
The functor $M \mapsto S^{-1}M$ is exact, so $S^{-1}R$ is a [[flat|73RYYEVDKA]] $R$-module.

Example: let $R = \Z$, $M = \Z/3\Z$ and $\mathfrak p = 2\Z$.
Then $3 \in S = \Z \setminus 2\Z$, and $m/s = 3m/(3s) = 0$ for every $m$, so $M_{(2)} = 0$.
For $\mathfrak p = 3\Z$, the elements of $S$ act invertibly on $\Z/3\Z$, and $M_{(3)} \cong \Z/3\Z$.

Non-example: the relation "$tm = sn$" without the factor $u$ is in general not transitive, as for the [[localization of a ring|4C1AE84N5N]], which is the case $M = R$.