\begin{definition}[Definition:Localization of Ring]
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: 
:325px
\end{definition}