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.

Also see

 * Localization of Ring Exists
 * Localization of Ring is Unique
 * Definition:Localization of Module
 * Definition:Localization of Algebra

Special cases

 * Definition:Localization of Ring at Element
 * Definition:Localization of Ring at Prime Ideal
 * Definition:Field of Fractions
 * Definition:Total Ring of Fractions