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Jordan mappings of semiprime rings. (English) Zbl 0691.16040

An additive mapping \(\theta\) of a ring R into a 2-torsion free ring \(R'\) is called a Jordan homomorphism if \(\theta (ab+ba)=\theta (a)\theta (b)+\theta (b)\theta (a)\) for all \(a,b\in R\). For \(R'\) prime, every Jordan onto homomorphism (\(\twoheadrightarrow\)) is either a homomorphism or an antihomomorphism, this is a well known result. For \(R'\) containing two disjoint ideals \(U'\) and \(V'\) and \(\phi:R\to U'\) a homomorphism and \(\psi:R\to V'\) an antihomomorphism, the mapping \(\theta =\phi +\psi\), which is a Jordan homomorphism is called a direct sum of \(\phi\) and \(\psi\) (*). It was shown by Baxter and Martindale that a Jordan homomorphism \(\theta:R\twoheadrightarrow R'\) for a semiprime \(R'\) is not necessarily a direct sum as in (*); but there always exists an essential ideal E of R such that the restriction of \(\theta\) to E is such a direct sum of \(\phi:E\to R'\) and \(\psi:E\to R'\). The author extends this to show that E can be so choosen as to be the sum of the ideals U and V of R, such that \(\phi\) vanishes on V and \(\psi\) vanishes on U and for each \(x\in R\), \(\theta (ux)=\theta (u)\theta (x)\) \(\forall u\in U\) and \(\theta (vx)=\theta (x)\theta (v)\) \(\forall v\in V\). This also answers the question of Baxter and Martindale “Whether there is a way to choose the ideal E so that \(\theta\) (E) is an associative subring of \(R'''\), in the affirmative showing in fact \(\theta\) (E) is the essential (associative) ideal of \(R'.\)
The later part of the work removes the restriction of the requirement of characteristic \(\neq 3\) on Herstein’s result that a Jordan triple homomorphism \([\theta (aba)=\theta (a)\theta (b)\theta (a)]\) \(\theta:R\twoheadrightarrow R'\) where \(R'\) prime with characteristic \(\neq 2\) and \(\neq 3\), is of the form \(\pm \phi\) for \(\phi\) a homomorphism or an antihomomorphism of R onto \(R'\).
Finally the author proves that every additive mapping d of a 2-torsion free semiprimitive ring R, which satisfies \(d(aba)=d(a)ba+ad(b)a+abd(a)\) \(\forall a,b\in R\), is in fact a derivation, generalizing well known results of Herstein on Jordan derivations, showing such a Jordan derivation of a 2-torsion free prime ring, is a derivation.
Reviewer: S.A.Huq

MSC:

16W20 Automorphisms and endomorphisms
16N60 Prime and semiprime associative rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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