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A note on derivations, additive subgroups, and Lie ideals of prime rings. (English) Zbl 1017.16028

The authors prove the following theorem for \(R\) a prime ring with \(\text{char }R\neq 2,3\). If \(L\) is a noncentral Lie ideal of \(R\) and \(D\) is a nonzero derivation of \(R\), then the additive subgroup of \(R\) generated by \(\{x^Dx-xx^D\mid x\in L\}\) must contain a noncentral Lie ideal of \(R\). The authors first prove a consequence of this result, namely that \([L^D,L]\) contains a noncentral Lie ideal of \(R\), to prove the theorem. The main theorem generalizes a number of others in the literature.

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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References:

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