Nikolaus Vonessen, Actions of Linearly Reductive Groups on PI-Algebras, Trans. Amer. Math. Soc. 335 (1993), 425-442.

Abstract:

Let G be a linearly reductive group acting rationally on a PI-algebra R.  We study the relationship between R and the fixed ring RG, generalizing earlier results obtained under the additional hypothesis that R is affine.