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Jordan derivations on prime rings. (English) Zbl 0634.16021

The authors give a brief proof of a theorem of I. N. Herstein [Proc. Am. Math. Soc. 8, 1104-1110 (1958; Zbl 0216.072)] which states that a Jordan derivation of a prime ring of characteristic not two is an ordinary derivation. The definition of a Jordan derivation is an additive map D of a ring R so that D(a \(2)=D(a)a+aD(a)\) for all \(a\in R\). There is a slight error in the proof given by the authors in that the last case they consider should be when \(ab=ba\) but \(a\not\in Z(R)\), instead of \(a\not\in Z(R)\) and \(b\in Z(R)\). The basic idea of the proof is the same as that given in [I. N. Herstein, Topics in Ring Theory. (1969; Zbl 0232.16001)], but unlike that proof, the method of the authors here does not seem to yield the generalization to the case of characteristic two when one also assumes that the Jordan derivation D satisfies \(D(aba)=D(a)ba+aD(b)a+abD(a)\).
Reviewer: C.Lanski

MSC:

16W20 Automorphisms and endomorphisms
16N60 Prime and semiprime associative rings
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References:

[1] DOI: 10.2307/2032688 · Zbl 0216.07202 · doi:10.2307/2032688
[2] Herstein, Topics in ring theory (1969) · Zbl 0232.16001
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