A note on the Jacobson radical of Ore extensions

چکیده مقاله

Let $R$ be a ring with a monomorphism $\alpha$ and an $\alpha$-derivation $\delta$. In this article,  we  give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring $R[x; \alpha,\delta ]$ is semiprimitive reduced if and only if $R$ is $\alpha$-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 $\alpha$ and $\delta$, 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.