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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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