Generalized derivations with nilpotent values on Lie ideals in semiprime rings (Q6600963)

Let \(R\) be a prime ring of characteristic different from \(2\), \(Q_r\) right Martindale quotient ring. The center of \(Q_r\) is denoted by \(C\) and 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 study the commutativity of generalized derivations on Lie ideals. More precisely, they prove that:\N\NTheorem: Let \(R\) be a prime ring of characteristic different from \(2\), \(Q_r\) its right Martindale quotient ring, \(C\) its extended centroid, \(L\) a non-central Lie ideal of \(R\), \(n \geq 1\) a fixed integer, \(F\) and \(G\) two generalized derivations of \(R\). If \((F(x y) - G(x)G(y))^{n} = 0\), for any \(x, y \in L\), then there exists \(\lambda \in C\) such that \(F(x) = \lambda^2x\) and \(G(x) = \lambda x\), for any \(x \in R\).\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)].