Jacobson radicals of Ore extensions

Let R be a ring, be an automorphism of R, and D be a -derivation on R. We will show that if R is an algebra over a field of characteristic 0 and D is q-skew, then J (Rx;, D) =I R+I₀ where I=\r R: rx J (Rx;, D) \ and I₀=\₈ ₁rᵢxⁱ: rᵢ I\. We will prove that J (Rx;, D) R is nil if is locally torsion and one of the following conditions is given: (1) R is a PI-ring, (2) R is an algebra over a field of characteristic p>0 and D is a locally nilpotent derivation such that D=D. This answers partially an open question by Greenfeld, Smoktunowicz and Ziembowski.