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The main goal of the paper is to establish the existence of tensor productdecompositions for those prime ideals P of the generic algebra A of quantum nby n matrices which are invariant under winding automorphisms of A. Morespecifically, every such P is the kernel of a map from A to (A^+/P^+) tensor(A^-/P^-) obtained by composing comultiplication, localization, and quotientmaps, where A^+ and A^- are special localized quotients of A while P^+ and P^-are prime ideals invariant under winding automorphisms. Further, the algebrasA^+ and A^-, which vary with P, can be chosen so that the correspondencesending (P^+,P^-) to P is a bijection. The main theorem is applied, in a sequelto this paper, to completely determine the winding-invariant prime ideals inthe generic quantum 3 by 3 matrix algebra.

PRIME IDEALS INVARIANT UNDER WINDING AUTOMORPHISMS IN QUANTUM MATRICES