Primitive Jordan pairs and triple systems
\textit{A. D'Amour} and \textit{K. McCrimmon} proved [J. Algebra 177, 199-239 (1995; Zbl 0852.17032)] that the local algebra at \(b\) of a \(b\)-primitive Jordan pair is primitive, and an analogous result for Jordan triple systems. In the paper under review the authors complete the above result giving the following local characterization of primitivity for Jordan pairs and triple systems: A Jordan pair \(V=(V^+,V^-)\) is primitive at \(b\in V^-\) if and only if \(V\) is strongly prime and the local algebra of \(V\) at \(b\) is primitive. A Jordan triple system \(T\) is primitive (at some element) if and only if \(T\) is strongly prime and there exists \(b\in T\) such that the local algebra of \(T\) at \(b\) is primitive. By using this local characterization of primitivity and results on primitivity in Jordan algebras [the authors and \textit{F. Montaner}, J. Algebra 172, 530-533 (1995; Zbl 0829.17027)], they also prove that primitivity is inherited by nonzero ideals of primitive Jordan pairs, and that primitivity is inherited by any prime Jordan pair having an ideal which is primitive. This, together with a Jordan characterization of primitivity for associative pairs, allows them to obtain a description of primitive Jordan pairs in the spirit of the Zelmanov classification for strongly prime Jordan pairs. An analogous description is also given for primitive Jordan triple systems which extends to general (quadratic Jordan triple systems) the central result of \textit{V. G. Skosyrskij} [Algebra Logic 32, No. 2, 96-111 (1993); translation from Algebra Logika 32, No. 2, 177-202 (1993; Zbl 0802.17025)] for linear Jordan triple systems. A final section is devoted to showing how *-tight associative envelopes of special primitive Jordan systems are *-primitive.
scientific article; zbMATH DE number 930162