Differential polynomial rings over rings satisfying a polynomial identity

Abstract

Let R be a ring satisfying a polynomial identity and let δ be a derivation of R. We show that if R is locally nilpotent then R[x;δ] is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski. As a consequence we have that if R is a unital PI algebra over a field of characteristic zero then the Jacobson radical of R[x;δ] is equal to N[x;δ], where N is the nil radical of R.