A noncommutative version of the Hilbert basis theorem is used to show that certain [Rscr ]-symmetric algebras S[Rscr ](V) are Noetherian. This result applies in particular to the coordinate ring of quantum matrices A[Rscr ](V) associated with an R-matrix [Rscr ] operating on the tensor square of a vector space V, to show that, under a natural set of hypotheses on [Rscr ], the algebra A[Rscr ](V) is Noetherian and its augmentation ideal has a polynormal set of generators. As a corollary we deduce that these properties hold for the generic quantized function algebras Rq [G] over any field of characteristic zero, for G an arbitrary connected, simply connected, semisimple group over [Copf ]. That Rq[G] is Noetherian recovers a result due to Joseph [10], with a different proof.