Trivial minimal ideals of Jordan systems
The authors review a previous result of their own [\textit{J. A. Anquela} and \textit{T. Cortés}, Invent. Math. 168, No. 1, 83--90 (2007; Zbl 1213.17031)] on the triviality of minimal ideals of Jordan systems. A problem that, as many other questions arising in Jordan theory, was originally inspired by an associative result. The original associative result is that of minimal ideals of an associative algebra to be either simple or to have zero multiplication, which holds as a consequence of Andrunakievich's Lemma. This question was originally posed by Zhevlakov in connection with the possibility that a Jordan analogue of Andrunakievich's Lemma, not known by that time, hold for Jordan algebras. Though Andrunakievich's Lemma proved to be false, using combinatorial arguments \textit{V. G. Skosyrskiĭ} provided an affirmative answer to Zhevlakov's question for linear Jordan algebras [Sib. Math. J. 29, No. 2, 283--293 (1988); translation from Sib. Mat. Zh. 29, No. 2(168), 154--166 (1988; Zbl 0662.17015)] as later did \textit{Yu. A. Medvedev} in [Sib. Mat. Zh. 28, No. 6(166), 81--89 (1987; Zbl 0638.17011)]. Both Skosyrskiĭ and Medvedev considered trivial ideals as those having zero square. Whereas Skosyrskiĭ's and Medvedev's proofs heavily depend on the linearity of the Jordan algebras, to study minimal ideals of quadratic Jordan algebras \textit{Ng Seong Nam} and \textit{K. McCrimmon} [Proc. Am. Math. Soc. 88, 579--583 (1983; Zbl 0519.17010)] chose a different approach based on structure theory. Nam and McCrimmon concluded that a minimal ideal of a quadratic Jordan algebra either is trivial or has no proper ideals invariant under all inner derivations induced from the algebra (i.e. is \({\mathcal D}\)-simple). Nonetheless Nam and McCrimmon defined an ideal of a quadratic Jordan algebra to be trivial if it has zero cube. Trivial ideals of Jordan systems (algebras, pairs or triple systems) are similarly defined by Anquela and Cortés in their paper cited above, where using a result proved by \textit{J. A. Anquela, T. Cortés} and \textit{E. García} in [Pure Appl. Algebra 172, No. 2--3, 119--137 (2002; Zbl 1032.17055)] stating that the heart of a nondegenerate Jordan quadratic algebra (triple system or pair) is simple when nonzero, completely solved Nam and McCrimmon problem relative to the triviality of minimal ideals of quadratic Jordan systems. However, up to this point, it still remained open the problem of providing an unified answer to Zhevlakov's question, also implying that result from Skosyrskii and Medvedev for linear Jordan algebras. In order to provide such a generalization Anquela, Cortés and McCrimmon introduce a new level of (intrinsic) triviality for ideals of quadratic Jordan systems. As a result, if an ideal is said to be trivial when it is trivial as a subsystem (basically when it has zero cube), it will be said to be \(J\)-trivial, where \(J\) denotes the Jordan system containing the ideal, when all monomials products on the enveloping system \(J\) containing at least two factors from the ideal vanish. Although it is not known if a trivial ideal contains a \(J\)-trivial ideal, using some easy instrumental lemmas involving the multiplication algebra of the Jordan system, this happens to be true for minimal ideals, yielding in particular that trivial (zero cube) minimal ideals have zero square and therefore allowing the authors to extend Anquela and Cortés result to include those proved by Skosyrskiĭ and Medvedev in the linear setting. As an application of the triviality of minimal ideals, Anquela, Cortés and McCrimmon end up considering the existence of nonzero nilpotent ideals in semiprime Jordan algebras, to prove that a Jordan algebra satisfying any finiteness condition implying the existence of minimal ideals has no nonzero nilpotent ideals if the algebra is semiprime and stating otherwise that the Baer radical \(B(J)\) of \(J\) is the smallest ideal of \(J\) providing a quotient free of nilpotent ideals.
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