Generalized derivations having the same power values with left multiplications. (Q2854024)

Let \(R\) be a prime ring with noncommutative Lie ideal \(L\), nonzero left ideal \(I\), maximal right ring of quotients \(U\), and extended centroid \(C\). The authors consider when a generalized derivation \(\delta\) of \(R\), extended to \(U\), satisfies \(\delta(x)^n=(ax)^n\) for a fixed positive integer \(n\) and a fixed \(a\in U\), for all \(x\in L\) or for all \(x\in I\).NEWLINENEWLINE The main results show first that for the \(x\in L\) case \(\delta(x)=\lambda ax\) for some \(\lambda\in C\) and all \(x\in R\). When \(\delta(x)^n=(ax)^n\) for all \(x\in I\) then for some \(b\in U\), \(\delta(x)=bx\) and either \(Ib=Ia=0\) or else \(b=\lambda a\) as above but with \(\lambda^n=1\).