Generalized derivations with periodic values on prime rings (Q6616660)

A ring \(R\) is said to be periodic if, for every \(x\in R\), there exist two distinct positive integers \(m\) and \(n\) such that \(x^m=x^n\).\N\NLet \(F: R\to R\) be an additive map on a ring \(R\). Then \(F\) is called a generalized derivation if \(F(xy)=F(x)y+xd(y)\) for all \(x, y \in R\), where \(d: R\to R\) is a derivation.\N\NIn this paper, the author studies generalized derivations with periodic values on prime rings and obtain the following result.\N\N\textbf{Theorem} Let \(n\geq 2\) be a fixed integer and \(R\) be a prime ring whose characteristic is either zero or \(p>0\) such that \(p(2^n-2)\). Let \(Q_r\) and \(C\) denote the right Martindale ring and extended centroid of \(R\), respectively. The ring of all \(2\times 2\) matrices over \(C\) is denoted by \(M_2(C)\). Suppose that \(F\) and \(G\) are two nonzero generalized derivations of \(R\) satisfying \[\big ( F(x)x-xG(x)\big )^n=F(x)x-xG(x)\] for all \(x\in [R, R]\), then one of the following statements holds:\N\begin{itemize}\N\item [1.] There exists \(a\in Q_r\) such that \(F(x)=xa\) and \(G(x)=ax\) for all \(x\in R\).\N\item [2.] \(R\subseteq M_2(C)\) and there exist \(a, c\in R\) such that \(F(x)=ax+xc\) and \(G(x)=cx+xa\) for all \(x\in R\).\N\item [3.] \(R\subseteq M_2(C)\) and there exist \(a, b, q\in R\) with \((a-q)^n=a-q\) such that \(F(x)=ax+xb\) and \(G(x)=bx+xq\) for all \(R\).\N\end{itemize}\NNote that, in the above theorem, the condition of the characteristic of ring \(R\) should be ``\(p\nmid (2^n-2)\)'' instead of ``\(p(2^n-2)\)''. However, from the proofs of this paper, this condition can be replaced by ``the characteristic of \(R\) is not \((2^n-2)\)''.\N\NFor the entire collection see [Zbl 1537.16001].