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Jordan derivations of prime rings. (English) Zbl 0216.07202
A Jordan derivation of an associative ring $$A$$ is a derivation for $$A^+$$, the Jordan ring obtained from $$A$$ by replacing its associative multiplication by $$a\circ b= ab+ba$$. It is proved that if $$A$$ is a prime ring of characteristic not two, then any Jordan derivation of $$A$$ is an ordinary (associative) derivation. For characteristic 2, the definition of Jordan derivation is modified to obeying the derivative rules for products $$a^2$$ and $$aba$$ of $$A$$ (this is equivalent to the previous definition if the characteristic is not 2). Then the same conclusion holds if $$A$$ is prime of characteristic 2, and $$A$$ not a commutative integral domain. Some further results on Jordan derivations from the symmetric elements $$S$$ of a ring $$R$$ with involution into $$R$$ were later proved by W. S. Martindale III [J. Algebra 5, 232–249 (1967; Zbl 0164.03601)]. He showed that such a Jordan derivation extends uniquely to an ordinary (associative) derivation of $$R$$ in two cases: (1) $$R$$ is prime, contains $$\tfrac 12$$ and $$R$$ has an invertible skew element in its center, and (2) $$R$$ is simple, of characteristic $$\neq 2$$ and $$R$$ possesses two non-zero orthogonal symmetric idempotents whose sum is 1.
Reviewer: Earl J. Taft

##### MSC:
 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings
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