An elemental characterization of strong primeness in Jordan systems

Anquela, JoséA.; Cortés, Teresa; Loos, Ottmar; McCrimmon, Kevin

doi:10.1016/0022-4049(95)00086-0

Cited by 26 publications

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“…However, the proof given in this paper cannot be considered an alternative to the proof by McCrimmon in [9]. Indeed the results of this paper are based on the results of [5], which depend on the inheritance of regularity by local algebras proved in [6]. These latter results are based on the results by McCrimmon [9].…”

Section: Introductionmentioning

confidence: 77%

“…The proof follows from the corresponding results on inheritance of regularity conditions by subquotients [3,5,6,8], once it is shown that V 0 and V 2 are isomorphic to certain subquotients of V .…”

Section: Introductionmentioning

confidence: 95%

“…(i) If V is nondegenerate, then T is nondegenerate using (2.4)(i). Conversely, if T is nondegenerate, then V is nondegenerate, as in the proof of 3.4 of [6], where only the fact that I + ∩ I − = 0 is used. (ii) If T is strongly prime, then V is nondegenerate by (i).…”

Section: Lemma Let T = T 2 ⊕ T 1 ⊕ T 0 Be a Peirce Grading Of A Jordmentioning

confidence: 99%

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Peirce gradings of Jordan systems

Anquela

Cortés

2002

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 95%

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Peirce gradings of Jordan systems

Anquela

Cortés

2002

Proc. Amer. Math. Soc.

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“…A particularly important example of special Jordan pairs is ample subpairs of an associative pair with involution w x 7, 1. 7 . Similar constructions lead to the notions of special Jordan triple systems and algebras, together with the particular cases of ample subw x spaces of associative triple systems and algebras with involution 2, 7 .…”

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Primitive Jordan Pairs and Triple Systems

Anquela

Contés

1996

Journal of Algebra

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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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“…For a ∈ R, the a-homotope of R, denoted by R a , is the associative algebra over the same linear structure as R with the new product x · a y = xay. It is readily seen that the set Ker a = x ∈ R axa = 0 is an ideal of R a , so we can consider the local algebra R a /Ker a , which will be denoted by R a .A number of properties of Jordan systems are inherited by their local algebras (see [DAMc,ACLMc,ACMo2,AC]) and also lifted from local algebras to the whole Jordan system. In particular, the local algebras of a nondegenerate Jordan algebra J are nondegenerate, and we have U a z ∈ ann J x ⇒z ∈ ann J a x (1.17) which follows from the nondegenerate annihilator conditions…”

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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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An elemental characterization of strong primeness in Jordan systems

Cited by 26 publications

References 5 publications

Peirce gradings of Jordan systems

Peirce gradings of Jordan systems

Primitive Jordan Pairs and Triple Systems

Goldie Theory for Jordan Algebras

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