Noether bound for invariants in relatively free algebras

Let RR be a weakly noetherian variety of unitary associative algebras (over a field K   of characteristic 0), i.e., every finitely generated algebra from RR satisfies the ascending chain condition for two-sided ideals. For a finite group G and a d-dimensional G-module V   denote by F(R,V)F(R,V) the relatively free algebra in RR of rank d freely generated by the vector space V  . It is proved that the subalgebra F(R,V)GF(R,V)G of G  -invariants is generated by elements of degree at most b(R,G)b(R,G) for some explicitly given number b(R,G)b(R,G) depending only on the variety RR and the group G (but not on V  ). This generalizes the classical result of Emmy Noether stating that the algebra of commutative polynomial invariants K[V]GK[V]G is generated by invariants of degree at most |G||G|.