𝐺-graded central polynomials and 𝐺-graded Posner’s theorem

Let F \mathbb {F} be a characteristic zero field, let G G be a residually finite group, and let W W be a G G -prime and polynomial identity F \mathbb {F} -algebra. By constructing G G -graded central polynomials for W W , we prove the G G -graded version of Posner’s theorem. More precisely, if S S denotes all nonzero degree e e central elements of W W , the algebra S − 1 W S^{-1}W is G G -graded simple and finite dimensional over its center. Furthermore, we show how to use this theorem in order to recapture a result of Aljadeff and Haile stating that two G G -simple algebras of finite dimension are isomorphix if and only if their ideals of graded identities coincide.