Nilpotent and invertible values in semiprime rings with generalized derivations

Let R be a semiprime ring and F be a generalized derivation of R and n >= 1 a fixed integer. In this paper we prove the following: (1) If (F(xy) - yx)(n) is either zero or invertible for all x, y is an element of R, then there exists a division ring D such that either R = D or R = M(2)(D), the 2 x 2 matrix ring. (2) If R is a prime ring and I is a nonzero right ideal of R such that (F(xy) - yx)(n) = 0 for all x, y is an element of I, then [I, I]I = 0, F(x) = ax + xb for a, b is an element of R and there exist alpha, beta is an element of C, the extended centroid of R, such that (a - alpha)I = 0 and (b - beta)I = 0, moreover ((a + b) x - x)I = 0 for all x is an element of I.