Identities with generalized derivations in prime rings. (Q1762387)

Let \(R\) be a prime ring with \(\text{char\,}R\neq 2\), \(Q_l\) and \(Q_r\) the left and right Martindale quotient rings of \(R\), and \(F,G\colon R\to R\) generalized derivations of \(R\). The authors prove that if for all \(x\in R\), \(F(x)G(x)=0\) then \(F(x)=xa\) for \(a\in Q_l\) and \(G(x)=bx\) for \(b\in Q_r\) with \(ab=0\). When \(F(x)G(x)+G(x)F(x)=0\) for all \(x\in R\) then either \(F=0\) or \(G=0\).