Differential polynomial rings over rings satisfying a polynomial identity

Abstract

Let $R$ be a ring satisfying a polynomial identity and let $\hat{I}´$ be a derivation of $R$. We show that if $N$ is the nil radical of $R$ then $\hat{I}´(N)\subseteq N$ and the Jacobson radical of $R[x;\hat{I}´]$ is equal to $N[x;\hat{I}´]$. As a consequence, we have that if $R$ is locally nilpotent then $R[x;\hat{I}´]$ is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski.

7 pages