Definition:Localization of Ring

Definition
Let $A$ be a commutative ring with unity.

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

Then a commutative ring with unity $A_S$ together with a ring homomorphism $\iota: A \to A_S$ is the localisation of $A$ at $S$ :


 * $(1): \quad \iota \left({S}\right) \subseteq A_S^\times$

where $A_S^\times$ is the group of units of $A_S$


 * $(2): \quad$ If:
 * $B$ is any ring
 * $g: A \to B$ is a ring homomorphism such that:
 * $g \left({S}\right) \subseteq B^\times$
 * then 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 localisation 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 localisation 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 localisation of $A$ at $S$.

Also see

 * Localisation of Ring Exists
 * Localisation of Ring is Unique