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 $$R$$.

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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