Definition:Localization of Module

Definition
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: Such that:
 * $M_S$ 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 $A$
 * 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.

Also see

 * Equivalence of Definitions of Localization of Module
 * Definition:Localization of Module Functor