Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:

An ideal $J\subset S$ is invariant under the action of the group $\mathrm{Hom}(B,K)$ if and only if $\eta_R(J)\subset J\Sigma$, where $\eta_R$ is the right unit.

Is it clear where the group action of $\mathrm{Hom}(B,K)$ on $S$ or $J$ comes from? Obviously, by the fact that the algebroid is split, there is an action of $\mathrm{Hom}(B,K)$ on $\mathrm{Hom}(S,K)$.