Generalized Derivations with Periodic Values on Prime Rings

Let R be a prime ring whose characteristic is either zero or \(p>0\) such that \(p(2^n-2)\) . Let \(Q_r\) be its right Martindale quotient ring, C its extended centroid and F and G non-zero generalized derivations of R. If \( \biggl (F(x)x-xG(x)\biggr )^n=\biggl (F(x)x-xG(x)\biggr ) \) for all \(x\in [R,R]\) , with \(n\ge 2\) fixed integer, then one of the following holds: