P.I. Algebras with Hopf Algebra Actions,

IfAis a p.i. algebra in characteristic zero with action from a finite-dimensional semisimple Hopf algebraH, thenAhas a nilpotentH-idealNsuch thatA/Nwill beH-verbally semiprime. EveryH-verbally semiprime algebra isH-p.i. equivalent to a direct sum ofH-verbally prime algebras. In the case of a finite group action or a grading by an abelian group, we show that the sum can be taken to be finite. In the case of an action by a finite cyclic groupG, we classify allG-p.i. algebras, up to equivalence. This paper generalizes the work of A. R. Kerner (1985,Math. USSR Izv.25).