Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is there a notion of an ideal class group of a Dedekind ring spectrum (Picard group)?

Thanks

PS This question was already asked on math.stackexchange, but I have heard there are some people on this site working on this sort of thing.