  The main goal of the paper is to establish the existence of tensor product
decompositions for those prime ideals P of the generic algebra A of quantum n
by n matrices which are invariant under winding automorphisms of A. More
specifically, every such P is the kernel of a map from A to (A^+/P^+) tensor
(A^-/P^-) obtained by composing comultiplication, localization, and quotient
maps, where A^+ and A^- are special localized quotients of A while P^+ and P^-
are prime ideals invariant under winding automorphisms. Further, the algebras
A^+ and A^-, which vary with P, can be chosen so that the correspondence
sending (P^+,P^-) to P is a bijection. The main theorem is applied, in a sequel
to this paper, to completely determine the winding-invariant prime ideals in
the generic quantum 3 by 3 matrix algebra.
