Let \(R\) be a prime ring with \(\text{char}(R)\neq 2\), and let \(U\) and \(C\) denote respectively the right Utumi quotient ring of \(R\) and the extended centroid of \(R\). Let \(f(x_1,\dots,x_n)\) be a polynomial over \(C\), and define \(f(R)\) to be \(\{f(r_1,r_2,\dots,r_n)\mid r_i\in R\}\). Let \(F\) be a non-zero generalized derivation on \(R\).
The authors show that if there exists a fixed \(k\geq 1\) and a noncentral Lie ideal \(L\) such that \([F(u),F(v)]_k=0\) for all \(u,v\in L\), or if \(f(x_1,\dots,x_n)\) is a noncentral polynomial and \([F(u),F(v)]=0\) for all \(u,v\in f(R)\), then \(R\) is an order in a central simple algebra of dimension at most 4 over its center and there exist \(a,b\in U\) such that \(a-b\in C\) and \(F(x)=ax+xb\) for all \(x,y\in R\).
Denoting the skew commutator \(xy+yx\) by \(x\circ y\) as usual, they show that if \(F(u)\circ F(v)=0\) for all \(u,v\in f(R)\), then \(f(x_1,\dots,x_n)\) is central valued on \(R\); and they also prove that if \(F(u)\circ F(v)=u\circ v\) for all \(u,v\in f(R)\), then either \(f(x_1,\dots,x_n)\) is central valued on \(R\) or there exists \(\alpha\in C\) such that \(F(x)=\alpha x\) and \(\alpha^2x=x\) for all \(x\in R\).