Definition 8.4.12. Let \(R\) be a ring spectrum and let \(S \subseteq \pi _*(R)\) be a multiplicative subset of homogeneous elements. Given a left \(R\)-module \(M\), its \(S\)-localization is an \(S\)-local left \(R\)-module \(S^{-1}M\) equipped with a morphism of left \(R\)-modules \(\eta \colon M \to S^{-1}M\) satisfying the property that for every other \(S\)-local left \(R\)-module \(N\), precomposition with \(\eta \) induces an equivalence of animae \[ - \circ \eta \colon \Hom _{\LMod _R}(S^{-1}M,N) \iso \Hom _{\LMod _R}(M,N). \]
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