Power-Central Values and Engel Conditions in Prime Rings with Generalized Skew Derivations

Let R be a prime ring of characteristic different from 2 with extended centroid C, n≥ 1 a fixed positive integer, F, G: R→ R two non-zero generalized skew derivations of R. (I)If (F(x) x) n∈ C for all x∈ R, then the following hold: (a)if F is an inner generalized skew derivation, then either R⊆ M2(C) or R is commutative;(b)if F is not an inner generalized skew derivation, then R is commutative.(II)If [F(x) x, G(y) y] n= 0 for all x, y∈ R, then R is commutative unless when char(R) = p> 0 , G is an inner generalized skew derivation and R⊆ M2(C). © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.