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Interesting "epimorphisms" of $E_\infty$-ring spectra

$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ Suppose $i:A\to B$ is a map of $E_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod_B\to\Mod_A$ by ...