Nilpotent and invertible values in semiprime rings with generalized derivations

Abstract

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 ${x,y\in R}$, then there exists a division ring D such that either R = D or R = M ₂(D), the 2 × 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 ${x,y \in I}$, then [I, I]I = 0, F(x) = ax + xb for ${a,b\in R}$ and there exist ${\alpha, \beta \in C}$, the extended centroid of R, such that (a − α)I = 0 and (b − β)I = 0, moreover ((a + b)x − x)I = 0 for all ${x\in I}$.