Locally nilpotent skew extensions of rings

Abstract

We extend existing results on locally nilpotent differential polynomial rings to skew extensions of rings. We prove that if G={σt}t∈T is a locally finite family of automorphisms of an algebra R, D={δt}t∈T is a family of skew derivations of R such that the prime radical P of R is strongly invariant under D, then the ideal P〈T,G,D〉⁎ of R〈T,G,D〉, generated by P, is locally nilpotent. We then apply this result to algebras with locally nilpotent derivations. We prove that any algebra R over a field of characteristic 0, having a surjective locally nilpotent derivation d with commutative kernel, and such that R is generated by ker⁡d2, has a locally nilpotent Jacobson radical.