A note on the Jacobson radical of Ore extensions

Let R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R[x;α,δ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results.