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 $\mathscr{G}=\{\sigma_t\}_{t\in T}$ is a locally finite family of automorphisms of an algebra $R$, $\mathscr{D}=\{\delta_t\}_{t\in T}$ is a family of skew derivations of $R$ such that the prime radical $P$ of $R$ is strongly invariant under $\mathscr{D}$, then the ideal $P\langle T,\mathscr{G},\mathscr{D}\rangle^*$ of $R\langle T,\mathscr{G},\mathscr{D}\rangle$, 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 $\ker d^2$, has a locally nilpotent Jacobson radical.