We give an elementary and self-contained proof of the theorem which says that for a semiprime ring commutativity, Lie-nilpotency, and nilpotency of the Lie ring of inner derivations are equivalent conditions
We examine when the nil and prime radicals of an algebra are stable under q-skew σ-derivations. We provide an example which shows that even if q is not a root of 1 or if δ and σ commute in characteristic 0, then the nil and prime radicals need not be δ-stable. However, when certain finiteness conditions are placed on δ or σ, then the nil and prime radicals are δ-stable.