Prime Jordan Algebras Satisfying Local Goldie Conditions

López, Antonio Fernández; Rus, Eulalia Garcí

doi:10.1006/jabr.1995.1165

Cited by 10 publications

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“…We remark here that the condition of (8.4) is most significant in Local Goldie Theory [FG1,FG2]. Since Jordan domains are strongly prime, we obtain as a consequence of (8.4) and (8.1) the following result.…”

Section: Fernández López García Rus and Montanermentioning

confidence: 70%

“…A nonempty subset S ⊆ J is called a monad if U s t and s 2 are in S, for all s t ∈ S (see [FG1]). This definition of monad is a little stronger than the one given by Zelmanov [Z2], where only the first condition is required.…”

Section: Basic Results On Ordersmentioning

confidence: 99%

“…These, in turn, were, at least partially, inspired by the work of these authors and of the third-named author of this paper on maximal modular inner ideals, and these were suggested by the arguments of [Z2, Z3]. There are, however, some differences in the approach followed here compared to the one used in [Z2,Z3,and FG1]. On one hand, dealing with the quadratic theory makes it more natural to consider only * -envelopes and raises the problem of handling ample subspaces instead of spaces of symmetric elements.…”

Section: Introductionmentioning

confidence: 87%

“…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 classical quotients in a way suitable for the absence of a unit element.In view of the current "state of the art," there are two questions that should be addressed to claim some completeness in the jordanification of Goldie's theory. On the one hand, modern Jordan theory has been made independent of the restrictions on the characteristic of the ring of scalars through the use of quadratic Jordan algebras, and therefore it is natural to ask for a quadratic generalization of Zelmanov's results.…”

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confidence: 97%

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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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Section: Fernández López García Rus and Montanermentioning

confidence: 70%

Section: Basic Results On Ordersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 87%

mentioning

confidence: 97%

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Goldie Theory for Jordan Algebras

2002

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“…As in the case of an alternative algebra, we have the following result whose proof will appear in [8], and which is a key tool in the new approach [8] to Zelmanov's Theorem for Goldie Jordan algebras [19], [20]. [5] a notion of local order in a Jordan algebra which need not have a unit and proved that a Jordan algebra J is a local order in a simple Jordan algebra Q with dcc on principal inner ideals but which is not a non-artinian quadratic factor if and only if J is a prime nondegenerate Jordan algebra satisfying local Goldie conditions. In a subsequent paper we extended this result to nondegenerate Jordan algebras.…”

Section: Uniform Ideals and Zelmanov's Theorem For Goldie Jordan Algementioning

confidence: 99%