Power centralizing semiderivations of Lie ideals in prime rings

Abstract

If a semiderivation $\mathscr{F}$ with associated automorphism $\xi$  is induced on a non-central Lie ideal $\mathscr{L}$ of $\mathfrak{A}$  such that \begin{align*} \left[\mathscr{F}(\eta), \eta \right]^{n}\in\mathcal{Z(R)}, \end{align*} where $n$ is a fixed positive integer, and $\eta\in\mathcal{L}$, then it has been proven that either \begin{align*} Char(\mathfrak{A}) =0 \end{align*} or \begin{align*} Char(\mathfrak{A})>n+1, \end{align*} then $\mathfrak{A}$ satisfies  a standard identity in $4$ variables usually denoted by $s_4$.