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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 \(x\in R\).

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)