Commutators and generalized derivations acting on Lie ideals in prime rings (Q6630770)

Let \(R\) be a prime ring of characteristic different from \(2\) and \(3\), \(U\) be the Utumi quotient ring. The center \(C= Z(U)\) is called the extended centroid of the ring \(R\). An additive mapping \(d:R\to R\) is said to be a derivation of \(R\) if \(d(xy)=d(x)y+xd(y)\) holds for all \(x,y\in R\). An additive mapping \(F:R\to R\) is called a generalized derivation of \(R\) if there exists a derivation \(d:R\to R\) such that \(F(xy)=F(x)y+xd(y)\) holds for all \(x,y\in R\).\N\NIn the paper under review, the authors studied the commutativity of generalized derivations on noncentral Lie ideals. More precisely, they prove that:\N\NTheorem: Let \(R\) be a prime ring of char \((R) \neq 2, 3\) and \(L\) a noncentral Lie ideal of \(R\). Let \(U\) be the Utumi quotient ring of \(R\) and \(C = Z(U)\) be the extended centroid of \(R\). Suppose that \(F, G, H\) are three generalized derivations of \(R\) such that \[[F(u), u]G(u) + u[H(u), u] = 0\] for all \(u \in L\). Then either \(R\) satisfies standard polynomial \(s_{4}(x_{1}, x_{2}, x_{3}, x_{4})\) or one of the following holds:\N\begin{itemize}\N\item[1.] There exist \(\alpha, \beta \in C\) such that \(F(x) = \alpha x\) and \(H(x) = \beta x\) for all \(x \in R\);\N\item[2.] There exists \(\beta \in C\) such that \(G(x) = 0\), \(H(x) = \beta x\) for all \(x \in R\);\N\item[3.] There exist \(a, b \in U\) and \(0 \neq \mu \in C\) such that \(F(x) = xa\), \(G(x) = \mu x\), \(H(x) = bx\) for all \(x \in R\) with \(\mu a + b \in C\).\N\end{itemize}\N\NThe result is obtained by using the theory of generalized polynomial identities [\textit{K. I. Beidar} et al., Rings with generalized identities. New York, NY: Marcel Dekker (1996; Zbl 0847.16001)] and the theory of differential identities [\textit{V. K. Kharchenko}, Algebra Logic 17, 155--168 (1979; Zbl 0423.16011); translation from Algebra Logika 17, 220--238 (1978)].