Locally nilpotent skew extensions of rings

We extend existing results on locally nilpotent differential polynomial rings to skew extensions of rings. We prove that if $\mathscr{G}={\sigma_t}{t\in T}$isalocallyfinitefamilyofautomorphismsofanalgebra$R$,$\mathscr{D}={\delta_t}{t\in T}$isafamilyofskewderivationsof$R$ such that the prime radical $P$ of $R$ is strongly invariant under $\mathcal{D}$, then the ideal $P⟨T,\mathcal{G},\mathcal{D}{⟩}^{\ast }$ of $R\langle T,\mathscr{G},\mathscr{D}\rangle$,generatedby$P$, 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.