We show that the braided tensor product algebra A1⊗A2
of two module algebras A1,A2 of a quasitriangular Hopf algebra H is
equal to the ordinary tensor product algebra of A1 with a subalgebra of
A1⊗A2 isomorphic to A2, provided there exists a
realization of H within A1. In other words, under this assumption we
construct a transformation of generators which `decouples' A1,A2 (i.e.
makes them commuting). We apply the theorem to the braided tensor product
algebras of two or more quantum group covariant quantum spaces, deformed
Heisenberg algebras and q-deformed fuzzy spheres.Comment: LaTex file, 29 page