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Commutators with power central values on a Lie ideal. (English) Zbl 1009.16034
Summary: Let $$R$$ be a prime ring of characteristic $$\neq 2$$ with a derivation $$d\neq 0$$, $$L$$ a noncentral Lie ideal of $$R$$ such that $$[d(u),u]^n$$ is central, for all $$u\in L$$. We prove that $$R$$ must satisfy $$s_4$$, the standard identity in $$4$$ variables. We also examine the case $$R$$ is a 2-torsion free semiprime ring and $$[d([x,y]),[x,y]]^n$$ is central, for all $$x,y\in R$$.

##### MSC:
 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16N60 Prime and semiprime associative rings 16R50 Other kinds of identities (generalized polynomial, rational, involution)
##### Keywords:
prime rings; derivations; Lie ideals; semiprime rings; commutators
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