A note on commutators with power central values on Lie ideals. (Q882734)

The authors eliminate the 2-torsion restrictions of \textit{L. Carini} and \textit{V. De Filippis} [Pac. J. Math. 193, No. 2, 269-278 (2000; Zbl 1009.16034)]. Assume that \(R\) is a prime ring, \(L\) is a noncentral Lie ideal of \(R\), \(D\neq 0\) is a derivation of \(R\), and \(n\) is a fixed positive integer. The first main theorem shows that when \([D(x),x]^n\) is central for all \(x\in L\), then \(R\) satisfies the standard identity \(S_4\). The second main result assumes that \(R\) is semiprime and \([D([x,y]),[x,y]]^n\) is central for all \(x,y\in R\), and concludes that there is a central idempotent \(e\in U\), the left Utumi quotient ring of \(R\), so that: the extension of \(D\) to \(U\) is zero on \(eU\), and \((1-e)U\) satisfies \(S_4\).