Study of commutator involving X-generalized skew derivations and annihilators in prime rings

Let [Formula: see text] be a prime ring of characteristic is not [Formula: see text], [Formula: see text] its right Martindale ring of quotient and [Formula: see text] be the extended centroid of [Formula: see text]. Suppose that [Formula: see text] is a noncentral multilinear polynomial over [Formula: see text] and [Formula: see text] with [Formula: see text]. When [Formula: see text] is a generalized skew derivation, then Yarbil et al. [Annihilators and centralizers of generalized skew derivations on multilinear polynomials, Beitr. Algebra Geom. 59(3) (2018) 573–595] studied the situation [Formula: see text] for all [Formula: see text] and then described the forms of the map [Formula: see text]. Recently, when [Formula: see text] is an [Formula: see text]-generalized skew derivation, then Prajapati and Gupta [On [Formula: see text]-generalized skew derivations having centralizing commutator in prime rings, Comm. Algebra 50(7) (2022) 2997–3006] studied [Formula: see text] for all [Formula: see text], and then described the forms of the map [Formula: see text]. In this paper, we study the above identity of Yarbil et al., that is, [Formula: see text] for all [Formula: see text], when [Formula: see text] is an [Formula: see text]-generalized skew derivation, and then we described all possible forms of the map.