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

Question

$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ 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 restriction of scalars.

A reasonable question is to ask when this is fully faithful; studying the counit of the restriction-extension of scalars adjunction, it's pretty easy to check that this is the case if and only if $B\otimes_A B\to B$ (the "multiplication" map) is an equivalence.

By studying its sections, if I'm not mistaken, one checks that this is the case if and only if the two inclusions $i_0,i_1: B\to B\otimes_A B$ are equivalent as maps of $E_\infty$-$A$-algebras.

For this it suffices that $A\to B$ be an "epimorphism" of $E_\infty$-$A$-algebras (and I think it's actually equivalent), that is, that $\map(B,-)\to \map(A,-)$ be an inclusion of components; since $\map(A,-) \simeq *$, this amounts to saying that $\map(B,C)$ is empty or contractible for all $C$.

For instance, this happens if $B$ is a localization of $A$ at a certain set of classes $S\subset \pi_*(A)$ (for instance $\mathbb{S\to Q, Z\to Q}, ku\to KU,$ etc.)

My question is:

Are there interesting cases where this happens but it's not a localization in the above sense ?

In the $1$-categorical case, this question was asked about epimorphisms of commutative rings (for which $\Mod_B\to \Mod_A$ is fully faithful if and only if $A\to B$ is an epimorphism), and there are examples that are neither quotients nor localizations.

Here, quotients usually do not satisfy this property, as "$x=0$" becomes additional structure (e.g. $\mathbb F_p\otimes_\mathbb Z\mathbb F_p \simeq \mathbb F_p[\epsilon], |\epsilon|=1$ as $E_1$-algebras), so it seems reasonable to ask what "epimorphisms" can look like in this setting.

Answer 1

If $A$ is an $E_\infty$ ring spectrum and $i : A \to B$ is any map of $A_\infty = E_1$ ring spectra such that the multiplication $\mu : B \wedge_A B^{op} \to B$ is an equivalence, then $B \simeq LA$ where $L$ is some smashing Bousfield localization on the category of $A$-modules. In particular, $B$ will be $E_\infty$ and $i$ is an $E_\infty$ map. Taking $A = S$ and $L = L_n$ to be the Bousfield localization with respect to the Johnson-Wilson theory $E(n)$, for $0 < n < \infty$, gives examples that are not given by algebraic localization at any set $S$ of classes in $\pi_*(A)$. The case $n=1$ corresponds to localization at ($p$-local) topological $K$-theory, with $B = L_1 S$ closely related to the image-of-$J$ spectrum. See Definition 1.18 of Ravenel's 1984 Amer. J. Math. paper for the notion of a smashing localization, and Proposition 9.3.3 in my AMS Memoir for the stated relation to "smashing maps".