  It is a fundamental result in commutative algebra and invariant theory that a
finitely generated graded module over a commutative finitely generated graded
algebra has rational Hilbert series, and consequently the Hilbert series of the
algebra of polynomial invariants of a group of linear transformations is
rational, whenever this algebra is finitely generated. This basic principle is
applied here to prove rationality of Hilbert series of algebras of invariants
that are neither commutative nor finitely generated. Our main focus is on
linear groups acting on certain factor algebras of the tensor algebra that
arise naturally in the theory of polynomial identities.
