An Engel condition with generalized derivations on Lie ideals. (Q934288)

Let \(R\) be a prime ring with extended centroid \(C\), Martindale quotient ring \(Q\), nonzero generalized derivation \(d\), noncentral Lie ideal \(L\), and nonzero right ideal \(T\). Set \([x,y]_1=xy-yx\) and for \(k>1\) let \([x,y]_k=[[x,y]_{k-1},y]_1\). The authors consider the situation when for a fixed \(k\), \([d(x),x]_k=0\) for all \(x\in L\) or all \(x=[u,v]_1\) for \(u,v\in T\). In the first case, either \(d(x)=cx\) for \(c\in C\), or else \(R\) satisfies the standard identity \(S_4\) and then \(\text{char\,}R=2\) or \(d(x)=ax+xb\) for \(a,b\in Q\) with \(a-b\in C\). In the case when \(x=[u,v]_1\) for \(u,v\in T\) then \(d(x)=cx\) for \(c\in R\) with \((c-z)I=0\) for \(z\in C\), or \(IC=eRC\) for \(e^2=e\in\text{soc}(RC)\) and \(eRCe\) satisfies \(S_4\). In the last case, \(\text{char\,}R=2\) or \(d(x)=ax+xb\) for some \(a,b\in R\) with \((a-b+z)I=0\) for \(z\in C\).