I was reading [this question](https://mathoverflow.net/questions/731/points-in-algebraic-geometry-why-shift-from-m-spec-to-spec) on why algebraic geometry looks at prime ideals instead of only maximal ideals, and I understand Anton's answer, but I'm a little confused as to how this fits with Hilbert's Nullstellensatz - affine algebraic sets are in bijection with radical ideals, not prime ideals, and it seems like we'd want the extra information we'd get by looking at "RadSpec(R)" (my own imagined notation). Also, the preimage of a radical ideal is radical, so there isn't the same objection as to maximal ideals - "RadSpec" would also be a contravariant functor.


So, why not radical ideals instead of only prime ideals, and what kinds of things could we say about RadSpec(R) even if they aren't very interesting?



MaxSpec, Spec, ... "RadSpec"? Or, why not look at all radical ideals?
The short answer is that every radical ideal is the intersection of the prime ideals containing it, so that the pullback map on Specs determines the pullback map on your wouldbe RadSpecs. 


Note that a similar reason underlies the success of looking only at maximal ideals in classical algebraic geometry: a finitely generated algebra over a field is a Hilbert-Jacobson ring: every radical ideal is the intersection of the *maximal* ideals containing it. 


I won't rule out the possibility that RadSpec could be useful for something, though...


