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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 $1.00+ $.25 per page

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