Some results for generalized Lie ideals in prime rings with derivation. II (Q2723215)

Let \(R\) be a prime ring, \(\text{char }R\neq 2\), \(Z\) the center of \(R\), \(D\) a nonzero derivation of \(R\), and \(\sigma,\tau\colon R\to R\). For \(x,y\in R\) set \([x,y]_{\sigma,\tau}=x\sigma(y)-\tau(y)x\), and for \(S,T\subseteq R\) let \([S,T]_{\sigma,\tau}\) be the additive subgroup of \(R\) generated by \(\{[s,t]_{\sigma,\tau}\mid s\in S\) and \(t\in T\}\). The authors claim the following: if \(a\in R\) so that \([D(R),a]_{\sigma,\tau}=0\) or \(D([R,a]_{\sigma,\tau})=0\) then \(\sigma(a)+\tau(a)\in Z\). In addition if \((x,y)_{\sigma,\tau}=x\sigma(y)+\tau(y)x\) and \(M\) is a nonzero ideal of \(R\), then \(([R,M]_{\sigma,\tau},a)_{\sigma,\tau}= 0\) forces \(a\in Z\). Finally, if \(I=\text{id}_R\) then \((D(R),a)_{I,I}=0\) if and only if \(D((R,a)_{I,I})=0\).