Eulalia García Rus

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.

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

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

Banach triples with generalized inverses

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AN EXTENSION OF THE ZEL'MANOV-GOLDIE THEOREM

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Goldie Theory for Jordan Algebras

Penrose showed in [6] that for every complex rectangular matrix a∈Mnxm(ℂ) there exists a unique b∈Mnxm(ℂ), the generalized inverse of a, satisfying the following four conditions: i) a a*b=a, ii) b a*a=a, iii) b b*a=b, iv) a b*b=b.These condition can be expressed in terms of the triple product =a b*c defined in A=Mnxm(ℂ), b* being the adjoint of b. With this triple product and with the operator norm, A is a complex Banach (associative) triple. In this note we will determine all the complex Banach triples with generalized inverses.

Prime associative triple systems with nonzero socle

A celebrated theorem for Jordan algebras due to Zel’manov [19, 20] states that a (linear) Jordan algebra J is an order in a semisimple artinian Jordan algebra Q if and only if J is semiprime, satisfies the annihilator chain condition, and does not contain infinite direct sums of inner ideals. Reciently [10], a notion of local order has been introduced in a (not necessarily unital) associative ring and proved a Goldie-like characterization of local orders in semiprime rings with dcc on principal one-sided ideals [11], equivalently, coinciding with their socles. Inspired by these ideas, we developed in [5, 6] a theory of local orders in Jordan algebras which need not have a unit and proved a natural extension of Zel’manov-Goldie theorem.