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Higher derivations and a theorem by Herstein. (English) Zbl 1009.16036
The authors extend results showing that Jordan or Jordan triple derivations are derivations to the case of higher derivations of a \(2\)-torsion-free semiprime ring \(R\). If \(D=(d_i)_N\) is a Jordan higher derivation (JHD) of \(R\) then it is a Jordan triple higher derivation (JTHD), and if \(D\) is JTHD of \(R\) then it is a higher derivation of \(R\). When \(U\) is a Lie ideal of \(R\) then \(D\) a JHD of \(U\) into \(R\) is a JTHD of \(U\) into \(R\). If \(R\) is a prime ring and \(u^2\in U\) for all \(u\in U\), then any JTHD of \(U\) into \(R\) is a higher derivation of \(U\) into \(R\).

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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