Miguel Gómez Lozano

Local rings of exchange rings

We characterize the exchange property for non-unital rings in terms of their local rings at elements,and we use this characterization to show that the exchange property is Morita invariant for idempotent rings.We also prove that every ring contains a greatest exchange idela(with respect to the inclusion).

Quotient rings and Fountain-Gould left orders by the local approach

We study Fountain-Gould left orders in semiprime rings coinciding with their socles by means of local rings at elements.

EXCHANGE MORITA RINGS

In this paper we characterize the largest exchange ideal of a ring R as the set of those elements x ∈ R such that the local ring of R at x is an exchange ring. We use this result to prove that if R and S are two rings for which there is a quasi-acceptable Morita context, then R is an exchange ring if and only if S is an exchange ring, extending an analogue result given previously by Ara and the second and third authors for idempotent rings. We introduce the notion of exchange associative pair and obtain some results connecting the exchange property and the possibility of lifting idempotents modulo left ideals. In particular we obtain that in any exchange ring, orthogonal von Neumann regular elements can be lifted modulo any one-sided ideal.

Nondegeneracy for Lie Triple Systems and Kantor Pairs

We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems, and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic 0 are nondegenerate. Departamento de Matematica Aplicada, Universidad Rey Juan Carlos, 28933 Mostoles (Madrid), Spain e-mail: esther.garcia@urjc.es Departamento de Algebra, Geometŕia y Topoloǵia, Universidad de Malaga, 29071 Malaga, Spain e-mail: magomez@agt.cie.uma.es Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5 e-mail: neher@uottawa.ca Received by the editors September 4, 2008. Published electronically March 5, 2011. The first author was partially supported by the MEC and Fondos FEDER MTM2007-62390 and MICINN-MTM2010-16153, by FMQ264, and by URJC-I3-2010/00075/001. The second author was partially supported by the MEC and Fondos FEDER MTM2007-61978 and MICINN-MTM2010-19482, by FMQ264 and FMQ3737, and by URJC-I3-2010/00075/001. The third author was partially supported by Natural Sciences and Engineering Research Council of Canada Discovery Grant #8836–2006. AMS subject classification: 17A40, 17B60, 17B99.

Inner ideal structure of nearly Artinian Lie algebras

In this paper we study the inner ideal structure of nondegenerate Lie algebras with essential socle, and characterize, in terms of the whole algebra, when the socle is Artinian.

LEFT QUOTIENT RINGS OF ALTERNATIVE RINGS

In this paper we develop a Fountain–Gould-like Goldie theory for alternative rings. We characterize alternative rings which are Fountain–Gould left orders in semiprime alternative rings coinciding with their socle, and those which are Fountain–Gould left orders in semiprime artinian alternative rings.

Center, Centroid, Extended Centroid, and Quotients of Jordan Systems

In this article we prove that the extended centroid of a nondegenerate Jordan system is isomorphic to the centroid (and to the center in the case of Jordan algebras) of its maximal Martindale-like system of quotients with respect to the filter of all essential ideals.

THE MAXIMAL LEFT QUOTIENT RINGS OF ALTERNATIVE RINGS

ABSTRACT We study the notion of a (general) left quotient ring of an alternative ring and show the existence of a maximal left quotient ring for every alternative ring that is a left quotient ring of itself.ABSTRACT We study the notion of a (general) left quotient ring of an alternative ring and show the existence of a maximal left quotient ring for every alternative ring that is a left quotient ring of itself.

3-Graded Lie Algebras with Jordan Finiteness Conditions

Abstract A notion of socle is introduced for 3-graded Lie algebras (over a ring of scalars Φ containing ) whose associated Jordan pairs are non-degenerate. The socle turns out to be a 3-graded ideal and is the sum of minimal 3-graded inner ideals each of which is a central extension of the TKK-algebra of a division Jordan pair. Non-degenerate 3-graded Lie algebras having a large socle are essentially determined by TKK-algebras of simple Jordan pairs with minimal inner ideals and their derivation algebras, which are also 3-graded. Classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space provide a source of examples of infinite dimensional strongly prime 3-graded Lie algebras with non-zero socle. Other examples can be found within the class of finitary simple Lie algebras

On Quotient Rings in Alternative Rings

We introduce a notion of left nonsingularity for alternative rings and prove that an alternative ring is left nonsingular if and only if every essential left ideal is dense, if and only if its maximal left quotient ring is von Neumann regular (a Johnson-like Theorem). Finally, we obtain a Gabriel-like Theorem for alternative rings.