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Let $\mathfrak{R}$ be a weakly noetherian variety of unitary associativealgebras (over a field $K$ of characteristic 0), i.e., every finitely generatedalgebra from $\mathfrak{R}$ satisfies the ascending chain condition fortwo-sided ideals. For a finite group $G$ and a $d$-dimensional $G$-module $V$denote by $F({\mathfrak R},V)$ the relatively free algebra in $\mathfrak{R}$ ofrank $d$ freely generated by the vector space $V$. It is proved that thesubalgebra $F({\mathfrak R},V)^G$ of $G$-invariants is generated by elements ofdegree at most $b(\mathfrak{R},G)$ for some explicitly given number$b(\mathfrak{R},G)$ depending only on the variety $\mathfrak{R}$ and the group$G$ (but not on $V$). This generalizes the classical result of Emmy Noetherstating that the algebra of commutative polynomial invariants $K[V]^G$ isgenerated by invariants of degree at most $\vert G\vert$.

Noether bound for invariants in relatively free algebras