It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under finite intersections, must be $Spec(R)$ for some ring $R$. Is there a canonical (or any) way of reconstructing $R$ from its collection of prime ideals and the Zariski topology thereupon? What is the relationship of the functor $Spec:Rng\to SpectralSpaces$ and the functor (if it exists) going the other way $SpectralSpaces\to Rng$?

Thanks!

anyfield is the one-point space. $\endgroup$ – user91132 Feb 16 '12 at 21:43