\begin{definition}[Definition:Localization of Module]
Let $A$ be a commutative ring with unity.
Let $S \subseteq A$ be a multiplicatively closed subset.
Let $A_S$ be the localization of $A$ at $S$.
Let $M$ be an $A$-module.
A '''localization of $M$ at $S$''' is a pair $(M_S, \iota)$ where:
*$M_S$ is an $A_S$-module
*$\iota : M \to M_S$ is the '''localization map''', an $A$-module homomorphism to the restriction of scalars of $M_S$ to $A$
Such that:
:For every $A_S$-module $N$ and $A$-module homomorphism $f: M \to \operatorname{res}_A^{A_S}N$ to the restriction of scalars to $A$, there exists a unique $A_S$-module homomorphism $g : M_S \to N$ such that $f = g \circ \iota$.
That is, the precomposition mapping between modules of homomorphisms:
:$\operatorname{Hom}_{A_S} (M_S, N) \overset {\iota^*} \to \operatorname{Hom}_A(M, N)$ is a bijection.
\end{definition}