Lie ideals and generalized derivations in semiprime rings. (Q2800496)

Let \(R\) be a 2-torsion free, semiprime ring with center \(Z(R)\), and noncentral Lie ideal \(L\). Denote by \(I\) the ideal of \(R\) generated by \([L,L]\) and let \((F,d)\) be the generalized derivation of \(R\) defined as \(F(x)=ax+d(x)\) for some \(a\in R\), \(d\in\text{Der}(R)\), and all \(x\in R\), and assume that \(d(L)\neq 0\). Consider the following sets in \(R\): \(d(R)[L,R]\), \([d(R),L]\), \([a,L]\), \(a[L,R]\), \(aI\), and \(d(I)\). The authors show that each of these sets is \(\{0\}\) when \(F([x,y])\in Z(R)\) for all \(x,y\in L\), or when \([F(x),x]\in Z(R)\) for all \(x\in L\). When \((G,g)\) is another generalized derivation, and if for all \(x,y\in L\), \(F([x,y])=[y,G(x)]\) then the authors state a result with conclusions much like those above.