Antonio Fernández López

On the socle of a noncommutative Jordan algebra

In this paper we study some questions related to the socle of a nondegenerate noncommutative Jordan algebra. First we show that elements of finite rank belong to the socle, and that every element in the socle is von Neumann regular and has finite spectrum. Next we show that for Jordan Banach algebras the socle coincides with the maximal von Neumann regular ideal. For a nondegenerate noncommutative Jordan algebra, the annihilator of its socle can be regarded as a radical which is, generally, larger than Jacobson radical. Moreover, a nondegenerate noncommutative Jordan algebra whose socle has zero annihilator is isomorphic to a subdirect sum of primitive algebras having nonzero socle (which were described in [4]). Finally, these results are specialized to the particular case of an alternative algebra.In this paper we study some questions related to the socle of a nondegenerate noncommutative Jordan algebra. First we show that elements of finite rank belong to the socle, and that every element in the socle is von Neumann regular and has finite spectrum. Next we show that for Jordan Banach algebras the socle coincides with the maximal von Neumann regular ideal. For a nondegenerate noncommutative Jordan algebra, the annihilator of its socle can be regarded as a radical which is, generally, larger than Jacobson radical. Moreover, a nondegenerate noncommutative Jordan algebra whose socle has zero annihilator is isomorphic to a subdirect sum of primitive algebras having nonzero socle (which were described in [4]). Finally, these results are specialized to the particular case of an alternative algebra.

Strongly prime alternative pairs with minimal inner ideals

In this paper, we use the known classification of the finite capacity simple alternative pairs and the version of the Litoff Theorem for Jordan pairs to describe all the strongly prime alternative pairs with nonzero socle. We study the inheritance of some properties (primeness, nondegenerancy,…) when passing from the original alternative pair to the symmetrized pair. Thus, we can apply Jordan theoretical results to the alternative case.

The Lie Inner Ideal Structure of Associative Rings Revisited

The aim of this note is to complete the description of the Lie inner ideal structure of simple Artinian rings with involution and of simple rings with involution and minimal one-sided ideals. Inner ideals are classified by adopting a Jordan approach based on the notion of a subquotient of an abelian inner ideal.

Towards a goldie theory for jordan pairs

A Goldie theory for Jordan pairs is started in this paper. We introduce a notion of order in linear Jordan pairs and study orders in nondegenerate linear Jordan pairs with descending chain condition on principal inner ideals.

Annihilators of elements of the socle of a jordan algebra

The double annihilator of any element in the socle of a nondegenerate quadratic Jordan algebra coincides with the principal inner ideal generated by this element. Since the socle satisfies dec on principal inner ideals, it also satisfies ace on annihilators of principal inner ideals.

π-Complemented Algebras Through Pseudocomplemented Lattices

For an ideal I of a nonassociative algebra A, the π-closure of I is defined by

Inner ideal structure of nearly Artinian Lie algebras

\overline{I} = {\rm Ann}({\rm Ann} (I))

A characterization of the elements of the socle of a Jordan algebra

, where Ann(I) denotes the annihilator of I, i.e., the largest ideal J of A such that IJ = JI = 0. An algebra A is said to be π-complemented if for every π-closed ideal U of A there exists a π-closed ideal V of A such that A = U ⊕ V. For instance, the centrally closed semiprime ring, and the AW∗-algebras (or more generally, boundedly centrally closed C∗-algebras) are π-complemented algebras. In this paper we develop a structure theory for π-complemented algebras by using and revisiting some results of the structure theory for pseudocomplemented lattices.

Strongly prime algebraic Lie PI-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.

Banach triples with generalized inverses

Let J be a nondegenerate Jordan algebra over a field K of characteristic not 2 . Here we prove that an element b 6 J is in the socle if and only if J satisfies dec on all principal inner ideals UyJ , y 6 Kb+ U¡,J . By using this result we show that the socle of a quadratic extension Jp of J coincides with the quadratic extension Soc(J)p of its socle. Throughout this paper J denotes a (linear) Jordan algebra over a field K of characteristic / 2. Our standard references for Jordan algebras are [6], [7], [11], For x,y G J we write their product by x • y. For x,y, z g J we write (1) LX:J-+J Lxy = x-y (2) UX:J^J Uxy = 2L2x-Lx2 (3) {xyz} = (Ux+_-Ux-U_)y (4) Bxyz = z-{xyz} + UxUyz. The Jordan algebra J is said to be nondegenerate if U = 0 implies x = 0. An inner ideal is a subspace I of J such that U, J c /. For any x, y in J we have the principal inner ideal UxJ, the inner ideal I(x) = Kx + UxJ generated by x , and the Bergmann inner ideal Bx J [7]. For nondegenerate J, the socle Soc(J) is defined to be the linear span of all minimal inner ideals of J ; Soc(J) = 0 if J does not contain any minimal inner ideal. By [10], if J contains minimal inner ideal then Soc(/) is a direct sum of simple ideals each of which contains a completely primitive idempotent e ( UeJ is a division Jordan algebra). An associative algebra A is semiprime iff the Jordan algebra A+ defined by the product x • y = \(xy + yx) is nondegenerate. For semiprime A , the (associative) socle of A coincides with the socle of the Jordan algebra A+ (see [3]). It is well known that an element a G A is in the socle iff A satisfies dec on all principal left ideals contained in Aa. In fact, A satisfies dec on all left ideals contained in Aa for every a G Soc(^). In the workshop on Jordan structures held at the University of Ottawa in 1986, McCrimmon settled the Received by the editors February 21, 1989 and, in revised form, April 19, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 17C10, 17C65, 16A34. ©1990 American Mathematical Society 0002-9939/90