## 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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### References:

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