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A note on commutators with power central values on Lie ideals. (English) Zbl 1119.16036
The authors eliminate the 2-torsion restrictions of L. Carini and 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 \(S_4\).

MSC:
16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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