On Continuous and Adjoint Morphisms Between Noncommutative Prime Spectra

We study topological properties of the correspondence of prime spectra associated to a noncommutative ring homomorphism R -> S. Our main result provides criteria for the adjointness of certain functors between the categories of Zariski closed subsets of Spec(R) and Spec(S); these functors arise naturally from restriction and extension of scalars. When R and S are left noetherian, adjointness occurs only for centralizing and ``nearly centralizing'' homomorphisms.