Miscellaneous problems 2 — Classify all possible Bousfield classes of E-infinity ring spectra.

Classify all possible Bousfield classes of E-infinity ring spectra. I know very little about this problem. Note that the Spanier-Whitehead dual of the suspension spectrum of a space X is an E-infinity ring spectrum, with multiplication dual to the diagonal map. But you are forgetting the disjoint basepoint!! When you add, as you must, a disjoint basepoint to X, you find that the E-infinity ring spectrum has the Bousfield class of the sphere. Also, I once heard somebody--Jim McClure?--say that if you start with, say, MU, and you mod out by p, you find that you must also mod out by all the higher v's too in order to get an E-infinity ring spectrum. Therefore, I conjecture that if E is an E-infinity ring spectrum that kills a nontrivial finite spectrum X, then E has the Bousfield class of E(n) for some n. (I must be p-local here). We know these Bousfield classes do occur, since Morava E-theory is an E-infinity ring spectrum. (Goerss-Hopkins).