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Generalized derivations and commutators with nilpotent values on Lie ideals. (English) Zbl 1133.16022
Summary: Let $$R$$ be a prime ring of characteristic $$\neq 2$$ with right quotient ring $$U$$ and extended centroid $$C$$, $$g\neq 0$$ a generalized derivation of $$R$$, $$L$$ a non-central Lie ideal of $$R$$ and $$n\geq 1$$ such that $$[g(u),u]^n=0$$, for all $$u\in L$$. We prove that there exists an element $$a\in C$$ such that $$g(x)=ax$$, for all $$x\in R$$, unless when $$R$$ satisfies $$s_4$$ and there exists an element $$b\in U$$ such that $$g(x)=bx+xb$$, for all $$x\in R$$.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 16R50 Other kinds of identities (generalized polynomial, rational, involution) 16U70 Center, normalizer (invariant elements) (associative rings and algebras)