Localization of a ring spectrum

Definition 8.4.8 (Localization of a ring spectrum). Let \(R\) be a ring spectrum and let \(S \subseteq \pi _*(R)\) be a multiplicative subset of homogeneous elements. The localization of \(R\) at \(S\), denoted \(R[S^{-1}]\), is a ring spectrum equipped with a map of ring spectra \(\eta \colon R \to R[S^{-1}]\) that satisfies the following property: for every ring spectrum \(A\), precomposition with \(\eta \) induces a monomorphism of animae \[ - \circ \eta \colon \Hom _{\Alg }(R[S^{-1}],A) \hookrightarrow \Hom _{\Alg }(R,A) \] whose image consists of those algebra maps \(f\colon R \to A\) for which \(\pi _*(f)\) carries every element of \(S\) to a unit of the graded ring \(\pi _*(A)\).