A Note on \(\textrm{b}\) -Generalized Skew Derivations on Prime Rings

Let \(\mathcal {R}\) be a prime ring with a characteristic not equal to 2. Let \(\mathcal {U}\) and \(\mathcal {C}\) denote its Utumi quotient ring and extended centroid, respectively. Consider a non-central multilinear polynomial \(\phi (\zeta _1, \ldots , \zeta _n)\) over \(\mathcal {C}\) , and let \(\textbf{G}\) be a \(\textrm{b}\) -generalized skew derivation of \(\mathcal {R}\) , satisfying the identity: \( p \phi (\zeta )\textbf{G}(\phi (\zeta ))= \textbf{G}(\phi (\zeta )^2),\ p\ne 2,\ \forall \ \zeta =(\zeta _1\ldots ,\zeta _n)\in \mathcal {R}^n.\) The purpose of this paper is to classify all potential forms of the \(\textrm{b}\) -generalized skew derivation \(\textbf{G}.\)