Let \(R\) be a prime ring with right Utumi quotient ring \(U\), extended centroid \(C\), nonzero right ideal \(I\), and nonzero generalized derivation \(D\). For \(x,y\in R\) let \(xy-yx=[x,y]=[x,y]_1\) and for \(k>1\) set \([x,y]_k=[[x,y]_{k-1},y]\).
The main result in the paper assumes that \([D(f(a_1,\dots,a_n)),f(a_1,\dots,a_n)]_k=0\) for a nonzero multilinear \(f(X)\in C\{x_1,\dots,x_n\}\), \(k\geq 1\) fixed, and all \(a_j\in I\). Then either there are \(a\in U\) and \(c\in C\) with \(D(x)=ax\) and \((a-c)I=0\), or else there is \(e^2=e\in\text{soc}(RC)\) so that \(IC=eRC\) and one of the following holds: i) \(f((eRCe)^n)\subseteq eC\) when \(\text{char\,}R=0\); ii) \(\text{char\,}R=p>0\) and some \(f(X)^{p^s}\) has all central values on \(eRCe\), or both \(\text{char\,}R=2\) and \(R\) embeds in \(M_2(F)\) for a field \(F\); iii) there are \(a,b\in U\), \(c\in C\) with \(D(x)=ax-xb\), \((a+b+c)e=0\), and \(f(X)^2\) has all central values on \(eRCe\).