It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these are the only spectra for which a Kunneth theorem holds on their cohomology theories, they are the only spectra over which every module is a free module, and these are all the minimal classes in the Bousfield lattice). Are other cases known in which all the fields are classified? For instance, if we localize at an arbitrary $E_\infty$-ring spectrum $R$, is there a way of determining which spectra satisfy any of the three criteria mentioned above? Must they be localizations of the "global fields" (by global of course I still mean $p$-local)?