The structure of primitive quadratic Jordan algebras

In this paper we give a characterization of primitivity of Jordan pairs and triple systems in terms of their local algebras. As a consequence of that local characterization we extend to Jordan pairs and triple systems most of the known results about primitive Jordan algebras. In particular, we describe primitive Jordan pairs and triple systems over an arbitrary ring of scalars in the sense of ''The Structure of Primitive Quadratic Jordan Algebras'' by J. A. Anquela, T. Cortes, and Ž .

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“…w Ž .x The elements in H H X eat from any position 1, Remark 3.14 3 . 5 Indeed, the same argument shows: Ž . follows from 0.…”

Section: ž B Bmentioning

confidence: 83%

“…translation with obvious changes of the corresponding result for Jordan w x w x pairs and triple systems given in Section 5 of 2 and Th. 5.5 of 3 .…”

Section: žmentioning

confidence: 99%

Primitive Jordan Pairs and Triple Systems

Anquela

Contés

1996

Journal of Algebra

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“…Strongly prime PI-algebras have been dealt with in [ACMo1], but here we are interested in nondegenerate algebras. We will need this more general case when studying the singular ideal in Section 6.…”

Section: Not So Basic Results On Ordersmentioning

confidence: 99%

Goldie Theory for Jordan Algebras

2002

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dedicated to professor holger petersson on the occasion of his 60th birthdayIt is shown that Zelmanov's version of Goldie's conditions still characterizes quadratic Jordan algebras having an artinian algebra of quotients which is nondegenerate. At the same time, Jordan versions of the main notions of the associative theory, such as those of the uniform ideal, uniform element, singular ideal, and uniform dimension, are studied. Moreover, it is proved that the nondegenerate unital Jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate Jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element.  2002 Elsevier Science (USA) Key Words: Jordan algebra; order; uniform element; singular ideal. INTRODUCTIONOne of the cornerstones of ring theory is Goldie's theorem characterizing those rings that have a nondegenerate artinian ring of quotients in 1 This work is supported by DGI Grants PB097-1069-C02-01 and PB097-1069-C02-02. 2 Deceased (October 5, 1999). fernández lópez, garcía rus, and montaner terms of ascending chain conditions [G1, G2, LC], thus linking for associative algebras two of the main notions of commutative ring theory. This provides the basic framework for the study of the Noetherian condition in associative ring theory, since it connects the study of that condition to what one should probably call the classical ring theory: rings with descending chain condition. Moreover, the kind of construction of quotients that is involved in those results is particularly well behaved since it, in addition to containing inverses for all regular elements (as any ring of quotients deserving its name should), consists of Ore's "one-sided" fractions of elements of the original ring. Further developments include a deepened study of the notions arising in those results, e.g., uniform ideals and nonsingularity.As for Jordan theory, it is natural to ask whether similar results can be achieved. The question was raised by Jacobson [J1, p. 426] in connection with the construction of (possibly exceptional) Jordan algebras as "Ore localizations" of Jordan domains. From a more structure-theoretic viewpoint, the early results of Britten [B1-B3] and Montgomery [Mon] deal with that question in the case of linear Jordan algebras J = H R * of symmetric elements of associative rings with involution. A general approach to localization of Jordan algebras was laid out by Jacobson et al. in [JMcP]. Based on the localization of the monoid of U-operators, they establish rather intricate "common multiple" conditions for the existence of localization of Jordan algebras. A definitive answer came with the papers of Zelmanov [Z2, Z3], where he establishes analogues of Goldie's theorems for linear Jordan algebras, making use of his fundamental results on structure theory rather than through the direct approach of [JMcP]. Further developments have been made by the first two authors in [FG1,FG2] dealing with the notion of local order, which generalizes cl...

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“…Thus, the inheritance of regularity to H 0 R * from the regularity of R in the case of algebras (see, for example, the proof of [8,Sect. 4.9]) always requires the consideration of the core ideals introduced in [15].…”

Section: Introductionmentioning

confidence: 99%

Herstein's Theorems and Simplicity of Hermitian Jordan Systems

2001

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dedicated to professor j. marshall osborn on the occasion of his retirementIn this paper we extend Herstein's first construction relating associative and Jordan ideals to pairs and triple systems. As a consequence we show that an associative pair or triple system is simple if and only if its Jordan symmetrization is simple. We also generalize Herstein's second construction to ample subsystems of associative algebras, pairs, and triple systems, which provides information on their simplicity when the associative structure is simple.  2001 Elsevier Science

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The structure of primitive quadratic Jordan algebras

Cited by 22 publications

References 20 publications

Primitive Jordan Pairs and Triple Systems

Primitive Jordan Pairs and Triple Systems

Goldie Theory for Jordan Algebras

Herstein's Theorems and Simplicity of Hermitian Jordan Systems

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