Teresa Cortés

An elemental characterization of strong primeness in Jordan systems

Abstract We give elemental characterizations of strong primeness for Jordan algebras, pairs and triple systems. We use our characterization to study the transfer of strong primeness between a Jordan system and its local algebras and subquotients.

Simplicity of the heart of a nondegenerate Jordan system

Abstract In this paper we prove that the heart of a nondegenerate Jordan system (algebra, triple system or pair) is either simple or zero. We also obtain Herstein type results relating the hearts of associative systems and those of their corresponding Jordan systems.

GROWING HEARTS IN ASSOCIATIVE SYSTEMS

We show that associative systems with a sufficiently good module structure imbed in a primitive system with simple primitive heart, spanned by the original system and the heart, so extending the results of J. Pure Appl. Algebra 181 (2003) 131–139 to systems over more general rings of scalars. We also study associative systems with involution.

Jordan cubes and associative powers

Abstract We study the relations between the powers of an associative algebra and the ideal generated by its Jordan cube. As a consequence, we describe the heart of an associative algebra for which every local algebra is simple in terms of associative powers. We also provide examples which show that the results obtained are optimal.

Commuting U -operators in Jordan algebras

For elements x;y in a non-degenerate non-unital Jordan algebra over a commutative ring, the relation x y = 0 is shown to imply that the U-operators of x and y commute: UxUy = UyUx. The proof rests on the Ze 0 lmanov-McCrimmon classication [15] of strongly prime quadratic Jordan algebras.

Imbedding Lie Algebras in Strongly Prime Algebras

We show that any Lie algebra with a sufficiently good module structure imbeds in a strongly prime Lie algebra with simple nondegenerate heart, spanned as a direct sum by the isomorphic image of the original algebra and the heart.

Peirce gradings of Jordan systems

In this paper we prove that the diagonal components Vo and V 2 of a Peirce grading of a Jordan pair or triple system V, inherit strong primeness, primitivity and simplicity from V.

Local nilpotency of the McCrimmon radical of a Jordan system

Using the fact that absolute zero divisors in Jordan pairs become Lie sandwiches of the corresponding Tits–Kantor–Koecher Lie algebras, we prove local nilpotency of the McCrimmon radical of a Jordan system (algebra, triple system, or pair) over an arbitrary ring of scalars. As an application, we show that simple Jordan systems are always nondegenerate.

Outer Inheritance of Simplicity in Jordan Systems

Abstract In this article, we show that outer ideals of simple Jordan systems have simple cube. In most cases we obtain that the outer ideals inherit simplicity and give precise descriptions in the cases in which simplicity is not obtained.

Left, Right, and Inner Socles of Associative Systems

We investigate the basic properties of the different socles that can be considered in not necessarily semiprime associative systems. Among other things, we show that the socle defined as the sum of minimal (or minimal and trivial) inner ideals is always an ideal. When trivial inner ideals are included, this inner socle contains the socles defined in terms of minimal left or right ideals.