Let R be a prime ring of characteristic different from 2 and 3, Qr its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y \in R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y \in R. We prove that, if F is a nonzero generalized skew derivation of R such that F(x)×[F(x), x]n = 0 for any x \in L, then either there exists λ \in C such that F(x) = λx for all x \in R, or R\subset M_{2}\left ( C \right ) and there exist a \in Qr and λ \in C such that F(x) = ax + xa + λx for any x \in R., Vincenzo De Filippis., and Obsahuje seznam literatury
Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R with the same associated automorphism α, and p(x1, ..., xn) is a non-central polynomial over C such that \left[ {F(x),\alpha (y)} \right] = G(\left[ {x,y} \right]). for all x,y\in \left \{ p\left ( r_{1},...,r_{n} \right ):r_{1},...,r_{n}\in R\right \}. The there exist \lambda \in C such that F(x) = G(x) = λα(x) for all X\in R., Vincenzo De Filippis., and Obsahuje seznam literatury