Centralizers satisfying polynomial identities

Abstract

The following results are proved: IfR is a simple ring with unit, and for someaeR witha n in the center ofR, anyn, such that the centralizer ofa inR satisfies a polynomial identity of degreem, thenR satisfies the standard identity of degreenm. WhenR is not simple,R will satisfy a power of the same standard identity, provided thata andn are invertible inR. These theorems are then applied to show that ifG is a finite solvable group of automorphisms of a ringR, and the fixed points ofG inR satisfy a polynomial identity, thenR satisfies a polynomial identity, providedR has characteristic 0 or characteristicp wherep✗|G|.