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Let $A$ be a ring, $S$ be a multiplicative subset, $M$ an $A$-module. let $\iota : A \to S^{-1}A$ be the map $a \mapsto a/1$. $\iota$ can be defined categorically as an initial object in the category of ring homomorphisms whose domain is $A$ which maps $S$ into the units (morphisms in this category are commuting triangles).

In Atiyah McDonald, as far as I can tell, there is no similar categorial definition for $S^{-1}M$. My question is: What are some good ways of defining $S^{-1}M$ categorically?

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Let ${}_A\mathrm{Mod}$ be the category of left $A$-modules. For all $M\in{}_A\mathrm{Mod}$, the localized module $M_S$ is in the full subcategory $\mathcal C$ of ${}_A\mathrm{Mod}$ spanned by all objects $N$ such that for all $s\in S$, the multiplication map $$m_{N,s}:n\in N\mapsto sn\in N$$ is isomorphism of abelian groups. Moreover, let $\iota:\mathcal C\to{}_A\mathrm{Mod}$ be the inclusion functor, One can show easily that the localization functor $(\mathord-)_S:{}_A\mathrm{Mod}\to\mathcal C$ is a left adjoint to $I$.

Since left adjoints are unique up to isomorphism, when then exist, this uniquely characterizes the localization functor $(\mathord-)_S$.

Finally, one should notice that a little extra work will show that the subcategory $\mathcal C$ I defined above is in fact naturally equivalent to the category ${}_{A_S}\mathrm{Mod}$ of left $A_S$-modules.

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