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Jordan mappings of semiprime rings. II. (English) Zbl 0727.16027
The author refines results on Jordan (triple) homomorphisms proved in an earlier paper [Part I, J. Algebra 127, 218-228 (1989; Zbl 0691.16040)]. An additive mapping of rings T: $R\to S$ satisfying $T(ab+ba)=T(a)T(b)+T(b)T(a)$ is a Jordan homomorphism, and is a Jordan triple homomorphism if $T(aba)=T(a)T(b)T(a)$. The results assume that S is 2-torsion free and that the annihilator of any ideal in S is a direct summand of S. The first theorem shows that when T is a Jordan homomorphism, then $R=I+J$ for ideals of R with $I\cap J=Ker T$, S is the direct sum of ideals T(I) and T(J), and T is an associative homomorphism on I and an antihomomorphism on J. For the second main theorem, $R\sp 2=R$ and T is a Jordan triple homomorphism. In this case there are four ideals $U\sb 1,...,U\sb 4$ of R whose sum is R and whose pairwise intersections are all Ker T, so that S is the direct sum of the ideals $T(U\sb i)$, and T acting on each $U\sb i$ is either an associative homomorphism, antihomomorphism, or the negative of one of these.

16W20Automorphisms and endomorphisms of associative rings
16N60Prime and semiprime associative rings
16W10Associative rings with involution, etc.
16D70Structure and classification of associative ring and algebras
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