Central polynomials for special Jordan rings without nilpotent elements, and an analog of Kaplansky's theorem

Abstract

In [7], a theory of nonassociative algebras with central polynomial was given, with several applications to Jordan algebras. The main theorem of this paper is that every special Jordan PI-ring J without nilpotent elements satisfies a class of central polynomials (related to the Formanek-Amitsur class of central polynomials for associative rings). This yields a “generic” Hamilton-Cayley equation for J. I f J is a torsion-free module over its center, then we can apply a result of Shirshov [9] (cf. [IS]) to [8], t o see that the “ring of central quotients” of J is simple and is either the Jordan algebra of a symmetric bilinear form or is finite dimensional (over its center), with a bound on the dimension computed in terms of the PI degree. In particular, every special Jordan PI-division ring is finite dimensional. This generalizes part of a result of Smith [ll], that every special algebraic simple Jordan PI-ring is finite dimensional. After this paper was distributed in preprint form, Bokut [17] announced the determination by Zelmanov of all Jordan division algebras of characteristic f2, thereby completing the Jacobson-Osborn structure theory of Jordan algebras with $ satisfying the descending chain condition on inner ideals; a generalization of Theorem 7 would follow immediately from Zelmanov’s results. To prove the main theorem we obtain a criterion for an element a of a special Jordan algebra without nilpotent elements to be central, namely, that [a, a, x] = 0 for all elements x. ([a, b, c] denotes the associator (ab)c a(&).) This is proved by verifying a fact about generalized identities of associative rings with involution. Other notational conventions are: [a, b] denotes ub bu, Z(R) denotes the center (of a ring R) = (a E R 1 [z, R, R] = [R, z, R] = [R, R, z] = [R, z] = 0). Nil ( ) denotes “the” maximal ideal of nilpotent elements, where an element Y is nilpotent if we can take some product of r, under suitable placement of paren-