Characteristics for $E_\infty$ ring spectra

Abstract: We introduce a notion of characteristic for connective $p$-local $E_\infty$ ring spectra and study some basic properties. Apart from examples already pointed out by Markus Szymik, we investigate some examples built from Hopf invariant $1$ elements in the stable homotopy groups of spheres and make some conjectures about spectra for which they may be characteristics; these appear to involve hard questions in stable homotopy theory.