Jordan derivations of prime rings. (English) Zbl 0216.07202

Herstein, I. N.

Proc. Am. Math. Soc. 8, 1104-1110 (1958).

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

Keywords:

prime ring of characteristic not two; Jordan derivation; ordinary associative derivation

Citations:

Zbl 0164.03601