Multilinear polynomial identities under the action of generalized skew derivations

Abstract

UDC 512.552

Suppose that $\mathcal{H}_1, \mathcal{H}_2$ and $\mathcal{H}_3$ are generalized skew derivations on prime ring $\mathcal{R}$ with ${\rm char}(\mathcal{R})\neq 2$ such that $\mathcal{H}_1\big(\mathcal{H}_2(\pi(q))\pi(q)\big)=\mathcal{H}_3(\pi(q)^2)$ for all $q=(q_1,\ldots,q_n) \in \mathcal{R}^n,$ where $\pi(q_1,\ldots,q_n)$ denotes the multilinear polynomial over $\mathcal{C},$ which is noncentral and nonidentity. We present all possible configurations of the maps $\mathcal{H}_1,  \mathcal{H}_2,$ and $\mathcal{H}_3.$ This extends the results obtained by V. de Filippis [Comm. Algebra, 49, No. 7, 2987–3009 (2021)].