Jordan algebras with minimum condition

Abstract

Let J be a Jordan algebra with minimum condition on quadratic ideals over a field of characteristic not 2. We construct a maximal nil ideal R of J such that J/R is a direct sum of a finite number of ideals each of which is a simple Jordan algebra. R must have finite dimension if it is nilpotent and this is shown to be the case whenever J has connected primitive orthogonal idempotents. Introduction. The first four sections of this paper deal with the construction of a maximal nil ideal for a Jordan algebra J with minimum condition on quadratic ideals over a field F of characteristic not 2. We call this ideal, R, the radical of J and show that J/R is semisimple in the sense of Jacobson [2]. We construct R by building up through the Peirce decomposition of J relative to a finite collection of primitive orthogonal idempotents. In order to facilitate computations, much of the construction is carried out under the assumption that J has an identity element. However, once R is constructed we show that this assumption is not really necessary. In the last section we deal with questions of the nilpotence of R and use the Coordinatization Theorem as our basic tool. We show that R is nilpotent if J has enough connected primitive orthogonal idempotents. Also if R is nilpotent, then it must be finite dimensional. 1. Preliminaries. In this paper an algebra will be an algebra over a field F of characteristic not 2, which is not necessarily associative or of finite dimension. J will always denote a commutative Jordan algebra. In order to make this paper self-contained we now recall some definitions and results from [2]. If x e J, U, denotes the linear operator 2RI RX2 on J, and if x, y, z E J then {xyz} will denote the trilinear product xy.z+yz.x-xz y. Ux = UXn uyUx = uxuyux zUx+y = zUx+zUy+2{xzy}. An element 0 =# b E J is called an absolute zero divisor of J if and only if JUb = 0. Presented to the Society January 22, 1970; received by the editors May 26, 1970. AMS 1969 subject classifications. Primary 1760.