Trivial minimal ideals of Jordan systems
type: Item
instance of: scholarly article
title: Trivial minimal ideals of Jordan systems
zbMATH Open document ID: 1278.17027
author: Teresa Cortés
author: Kevin McCrimmon
author: José A. Anquela
published in: Journal of Algebra
publication date: 2011-05-16T00:00:00Z
full work available at URL: http://hdl.handle.net/10651/6213
review text: 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 r…
reviewed by: Irene Paniello
Mathematics Subject Classification ID: 17C10
Mathematics Subject Classification ID: 17C17
Mathematics Subject Classification ID: 17C20
zbMATH DE Number: 5888487
zbMATH Keywords: Jordan system
zbMATH Keywords: minimal ideal
zbMATH Keywords: trivial ideal
MaRDI profile type: Publication
OpenAlex ID: W1989904284
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DOI: 10.1016/J.JALGEBRA.2010.10.016
description: scientific article; zbMATH DE number 5888487