  In this paper we study contravariant functors from the category of rings to
the category of sets whose restriction to the full subcategory of commutative
rings is isomorphic to the prime spectrum functor Spec. The main result reveals
a common characteristic of these functors: every such functor assigns the empty
set to M_n(C) for n >= 3. The proof relies, in part, on the Kochen-Specker
Theorem of quantum mechanics. The analogous result for noncommutative
extensions of the Gelfand spectrum functor for C*-algebras is also proved.
