Noether bound for invariants in relatively free algebras

Let R be a weakly noetherian variety of unitary associative algebras (over a field K of characteristic 0), i.e., every finitely generated algebra from R satisfies the ascending chain condition for two-sided ideals. For a finite group G and a d-dimensional (G) over dot-module V denote by F(R,V) the relatively free algebra in R of rank d freely generated by the vector space V. It is proved that the subalgebra F(R,V)(G) of G-invariants is generated by elements of degree at most b(R, G) for some explicitly given number b(R, G) depending only on the variety R 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](G) is generated by invariants of degree at most vertical bar G vertical bar. (C) 2016 Elsevier Inc. All rights reserved.