Left annihilator of generalized derivations on Lie ideals in prime rings. (Q2517008)

For an associative ring \(R\), if \(d\in\text{der}(R)\) and \(F\colon R\to R\) is additive, so that for all \(x,y \in R\) \(F(xy)=F(x)y+xd(y)\) then \((F,d)\) is called a generalized derivation of \(R\). The authors assume that \(R\) is a prime ring, \(a\in R\), \(L\) is a noncentral Lie ideal of \(R\), and \((F,d)\) is a generalized derivation of \(R\). The main result of the paper assumes that \(n,k>0\), \(m_1,\ldots,m_k\geq 0\) are fixed integers, with some \(m_j>0\) and proves that if \(a(d(x)^{m_1}F(x)^{m_2}d(x)^{m_3}F(x)^{m_4}\cdots)^n\) is an identity for \(L\), then \(a=0\) or \(R\) embeds in \(M_2(C)\) for a field \(C\).