  We initiate a unified, axiomatic study of noncommutative algebras R whose
prime spectra are, in a natural way, finite unions of commutative noetherian
spectra. Our results illustrate how these commutative spectra can be
functorially ``sewn together'' to form Spec R. In particular, we construct a
bimodule-determined functor Mod Z -> Mod R, for a suitable commutative
noetherian ring Z, from which there follows a finite-to-one, continous
surjection Spec Z -> Spec R. Algebras satisfying the given axiomatic framework
include PI algebras finitely generated over fields, noetherian PI algebras,
enveloping algebras of complex finite dimensional solvable Lie algebras,
standard generic quantum semisimple Lie groups, quantum affine spaces,
quantized Weyl algebras, and standard generic quantizations of the coordinate
ring of nxn matrices. In all of these examples (except for the
non-finitely-generated noetherian PI algebras), Z is finitely generated over a
field, and the constructed map of spectra restricts to a surjection Max Z ->
Prim R.
