Identities with product of generalized derivations of prime rings. (Q2862508)

Let \(F\) and \(G\) be two nonzero generalized derivations of a noncommutative prime ring \(R\), \(\text{char}(R)\neq 0\), with extended centroid \(C\) and let \(f(x_1,\ldots,x_n)\) be a multilinear polynomial over \(C\) which is not an identity for \(R\). The authors study the condition \(F(u)G(u)=0\), \(u\in f(R)\), where \(f(R)=\{f(r_1,\ldots,r_n):r_i\in R\}\).