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 $(A_S, \iota)$ where:
 * $A_S$ is a commutative ring with unity
 * $\iota: A \to A_S$ is a ring homomorphism

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

That is, the following diagram commutes:


 * LocalisationComdiag.png

Notation
The localization of $A$ at $S$ can be written $S^{-1} A$, or $A \left[{S^{-1} }\right]$.

If $\mathfrak p$ is a prime ideal of $A$, then by definition, $S = A \mathrel \backslash \mathfrak p$ is multiplicatively closed.

In this case is conventional to write $A_{\mathfrak p}$ for the localization of $A$ at $S$.

If $f \in A$ is some element, then $S = \left\{ {f^n: n \ge 0}\right\}$ is trivially multiplicatively closed, and it is common to write $A_f$ for the localization of $A$ at $S$.

Also see

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