Local inheritance in Jordan algebras

Anquela, José A.; Cortés, Teresa; Montaner, Fernando

doi:10.1007/bf01197216

Cited by 18 publications

(16 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…in general P ⊆ ann J ann J P , and the reverse inclusion follows from U ann J P ann J ann J P = 0 ⊆ P ⇒ ann J ann J P ⊆ P by the primeness of P, since ann J P is not contained in P. Thus J = J/P is strongly prime, and by [ACMo2] its local algebra Jā is also strongly prime. Since P ⊆ Ker J a Jā is isomorphic to J a , hence the local algebra of J at a is strongly prime.…”

Section: Proof (I) ⇒ (Ii) We Associate To Each Chainmentioning

confidence: 83%

“…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…”

mentioning

confidence: 99%

See 1 more Smart Citation

Goldie Theory for Jordan Algebras

2002

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Proof (I) ⇒ (Ii) We Associate To Each Chainmentioning

confidence: 83%

mentioning

confidence: 99%

Goldie Theory for Jordan Algebras

2002

Self Cite

View full text Add to dashboard Cite

show abstract

“…Unlike the algebra case, primitivity of a Jordan pair is considered at a particular element, and shown to be equivalent to primitivity of its local algebra at this element, under the assumption of strong primeness [4]. As a consequence, the results on local inheritance of primitivity in Jordan algebras [9] were an important tool to have some possibility of changing the element at which primitivity holds, which is essential in the description of primitive Jordan pairs and triple systems given in [4].In this paper we show the inheritance of primitivity by a strongly prime Jordan algebra from any primitive local algebra. Apart from its intrinsic interest, this result will play an important role in the definite solution of the above problem on the ªlocationº of primitivity for Jordan systems [5], as one could expect.…”

mentioning

confidence: 99%

“…As a consequence, the results on local inheritance of primitivity in Jordan algebras [9] were an important tool to have some possibility of changing the element at which primitivity holds, which is essential in the description of primitive Jordan pairs and triple systems given in [4].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Local-to-global inheritance of primitivity in Jordan algebras

Anquela

Cortés

1998

Archiv der Mathematik

Self Cite

View full text Add to dashboard Cite

We show that a strongly prime Jordan algebra having a primitive local algebra is primitive. As a tool, similar results concerning one-sided primitivity and Ã-primitivity of associative algebras are established.Introduction. The notion of local algebra of a Jordan system, introduced in [17], has proved to be a basic tool in the study of primitivity [3,4,7]. Unlike the algebra case, primitivity of a Jordan pair is considered at a particular element, and shown to be equivalent to primitivity of its local algebra at this element, under the assumption of strong primeness [4]. As a consequence, the results on local inheritance of primitivity in Jordan algebras [9] were an important tool to have some possibility of changing the element at which primitivity holds, which is essential in the description of primitive Jordan pairs and triple systems given in [4].In this paper we show the inheritance of primitivity by a strongly prime Jordan algebra from any primitive local algebra. Apart from its intrinsic interest, this result will play an important role in the definite solution of the above problem on the ªlocationº of primitivity for Jordan systems [5], as one could expect.The main ingredients of the proofs are the general properties of primitivity and its local inheritance in Jordan algebras [8,9], as well as some facts concerning primitive pairs and triple systems [3,4]. After a preliminary section, collecting known results which will be used in the sequel, we begin in Section 1 with the study of local-to-global inheritance of one-sided primitivity and Ã-primitivity in associative algebras. The corresponding result for Jordan algebras splits into two cases: the hermitian case and the ªfiniteº case. Due to the fact that the initial object is the local algebra, the above two cases are those of non-homotope-PI and homotope-PI Jordan algebras. This forces us to use the theory of pairs and triple systems, so that results on the relationship of a Jordan algebra and its underlying triple system are needed. Non-homotope-PI Jordan algebras are studied in Section 2 using the main result of Section 1, while the homotope-PI case in Section 3 follows directly by an argument involving Peirce decompositions and the socle. We unify the different cases in the main theorem of Section 4, getting also an important consequence on inheritance of primitivity from Peirce components.

show abstract

Primitivity in Jordan Systems is Ubiquitous

Anquela

Cortés

1998

Journal of Algebra

View full text Add to dashboard Cite

The main result of the paper asserts that if a Jordan pairX .-primitive at some 0 / b g V , then it is " -primitive at any 0 / b g V . Also, if a Jordan triple system T is primitive at some 0 / b g T, then it is primitive at any 0 / b X g T. As a tool, similar results concerning one-sided primitivity and )-primitivity of associative pairs and triple systems are established. Equivalences between our definitions of primitivity and some other appearing in the literature are also obtained as a consequence.

show abstract

Local inheritance in Jordan algebras

Cited by 18 publications

References 8 publications

Goldie Theory for Jordan Algebras

Goldie Theory for Jordan Algebras

Local-to-global inheritance of primitivity in Jordan algebras

Primitivity in Jordan Systems is Ubiquitous

Contact Info

Product

Resources

About