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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)
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