Local PI Theory of Jordan Systems II

Montaner, Fernando

doi:10.1006/jabr.2001.8779

Cited by 21 publications

(33 citation statements)

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“…As a consequence of that theorem we can improve the Jordan analogues of Kaplansky's and Posner-Rowen theorems obtained in [20][21][22].…”

Section: 7mentioning

confidence: 74%

“…Recall that U A is a unital associative algebra with two idempotents e 1 + e 2 = 1 such that if we consider the Peirce decomposition of U A with respect to these idempotents, then A = ((U A ) 12 , (U A ) 21 ) with the usual triple product. Every involution of A extends uniquely to an algebra involution of U A that satisfies e * 1 = e 2 [9, 3.2].…”

Section: 3mentioning

confidence: 99%

“…We refer to [21] for the related notion of extended centroid of a Jordan system J, which we denote C(J), and the attached scalar extension (for a nondegenerate J): its extended central closure C(J)J.…”

Section: 9mentioning

confidence: 99%

“…The only difference is that the role played by polynomial identities is now played by homotope polynomial identities (see [6-8, 20, 21]), that is, polynomial identities that hold in all homotope algebras of the system. This partially motivated the study of "local PItheory" in [20][21][22], that is, the study of Jordan systems with local algebras satisfying a polynomial identity, and in particular, of Jordan systems satisfying a homotope polynomial identity.…”

mentioning

confidence: 99%

“…The question of whether a theory of general identities of Jordan systems could be developed remained however open, and was the content of a question raised in [21], namely: does every PI Jordan system satisfy a homotope polynomial identity?…”

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confidence: 99%

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On Polynomial Identities in Associative and Jordan Pairs

Montaner

Paniello

2009

Algebr Represent Theor

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“…As a consequence of that theorem we can improve the Jordan analogues of Kaplansky's and Posner-Rowen theorems obtained in [20][21][22].…”

Section: 7mentioning

confidence: 74%

Section: 3mentioning

confidence: 99%

Section: 9mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On Polynomial Identities in Associative and Jordan Pairs

Montaner

Paniello

2009

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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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On Local Algebras of Maximal Algebras of Jordan Quotients

Montaner

Paniello

2020

Recent Advances in Pure and Applied Mathematics

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Local PI Theory of Jordan Systems II

Cited by 21 publications

References 22 publications

On Polynomial Identities in Associative and Jordan Pairs

On Polynomial Identities in Associative and Jordan Pairs

Goldie Theory for Jordan Algebras

On Local Algebras of Maximal Algebras of Jordan Quotients

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