Lemma 10.9.5. Let $R$ be a ring. Let $S \subset R$ be a multiplicative subset. The category of $S^{-1}R$-modules is equivalent to the category of $R$-modules $N$ with the property that every $s \in S$ acts as an automorphism on $N$.

Proof. The functor which defines the equivalence associates to an $S^{-1}R$-module $M$ the same module but now viewed as an $R$-module via the localization map $R \to S^{-1}R$. Conversely, if $N$ is an $R$-module, such that every $s \in S$ acts via an automorphism $s_ N$, then we can think of $N$ as an $S^{-1}R$-module by letting $x/s$ act via $x_ N \circ s_ N^{-1}$. We omit the verification that these two functors are quasi-inverse to each other. $\square$

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