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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 which is somehow "over" either $H\mathbb{Z}$ or $\mathbb{S}$? Additionally, is there a notion of an ideal class group of a Dedekind ring spectrum?

Thanks

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    @AaronMazel-Gee Algebraic geometers (working in the opposite category, of affine schemes) are to blame here ;-) And yes, S is initial in the category of *ring* spectra (it's analogue of $\mathbb Z$, not of a point).2011-12-05

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