# Definition:Localization of Ring

## Definition

Let $A$ be a commutative ring with unity.

Let $S \subseteq A$ be a multiplicatively closed subset of $A$.

A localization of $A$ at $S$ is a pair $\struct {A_S, \iota}$ where:

$A_S$ is a commutative ring with unity, the actual localization
$\iota: A \to A_S$ is a ring homomorphism, the localization homomorphism

such that:

$(1): \quad \map \iota S \subseteq A_S^\times$, where $A_S^\times$ is the group of units of $A_S$
$(2): \quad$ For every pair $\tuple {B, g}$ where:
$B$ is any ring with unity
$g: A \to B$ is a ring homomorphism such that $\map g S \subseteq B^\times$
there exists a unique ring homomorphism $h: A_S \to B$ such that:
$g = h \circ \iota$

That is, the following diagram commutes:

$\begin{xy}\xymatrix@L+2mu@+1em { A \ar[drdr]_*{g} \ar[rr]^*{\iota} & & A_S \ar[dd]^*{\exists ! h} \\ \\ & & B }\end{xy}$

## Notation

The localization of $A$ at $S$ can be written $S^{-1} A$, or $A \sqbrk {S^{-1} }$.

The notation $A_{\mathfrak p}$ is seen for the localization at a prime ideal $\mathfrak p$.

The notation $A_f$ is seen for the localization at an element $f \in A$.

## Also known as

A localization of a ring is also known as a ring of fractions.

## Linguistic Note

The word localization is spelt localisation in non-US English.