Jordan algebras of capacity two.

A recent paper by N. Jacobson' develops a theory of Jordan algebras with minimum condition which is similar to the associative theory of semisimple Artinian rings. Jacobson shows that any algebra satisfying his axioms is a direct sum of a finite number of simple algebras satisfying his axioms, and he deduces the structure of those simple algebras which contain three mutually orthogonal idempotents. The purpose of the present paper is to deduce the structure of simple algebras a satisfying his axioms which contain two (but not three) orthogonal idempotents. A Jordan algebra 3 with unity element 1 is called a division algebra if for every nonzero element x F 3 there is an element x-1 E 3 such that x x-l = 1 and x2 x-1 = x. We shall say that a Jordan algebra a with unity element 1 has capacity 2 if a contains two orthogonal idempotents el and e2 such that el + e2 = 1 and such that the subalgebras a, = {x F J~xei = x} are division algebras fori = 1,2. If Z is an associative algebra with unity element and with an involution x x and if 'y is a 2 X 2 matrix over 2, let '(2,-y) denote that Jordan subalgebra of 2+consisting of all 2 X 2 matrices A over i which satisfy A'y = 'yA, where A' denotes the conjugate transpose of A. Our main result is THEOREM 1. Let a be a simple Jordan algebra of capacity 2 over afield of characteristic : 2. Then either a is isomorphic to the Jordan algebra of a nondegenerate bilinearform on a vector space over an extension field of cf, or 3 is isomorphic to (Z2,Y) where -y E Z2 is invertible and diagonal, and where Z is either an associative division algebra with involution or else the direct sum of two associative division algebras which are interchanged by the involution. Remark: It is not difficult to see that a Jordan ring of capacity 2 is necessarily a Jordan algebra of capacity 2 over an appropriate field, so that there is no loss of generality in assuming operators from a field. At the end of this paper we state a generalization of Theorem 1 which weakens the hypothesis of simplicity. We also give there a related theorem on associative rings with involution which generalizes a previous result of ours.2 The proofs of these two theorems are similar to the proofs of the theorems that they generalize and are omitted here to save space. 1. Definitions and Preliminary Results.-Throughout this section and the next, a shall denote a simple Jordan algebra of capacity 2 over a field c1 of characteristic $ 2, and el and e2 shall be fixed orthogonal idempotents of a such that el + e2 = 1 and such that 1 = {x F Jfx-el = x} and 32 = {x F 3jx.e2 = x} are division algebras. Elements of S with subscripts 1,2, or 12 will be assumed to be elements of 3,, or 312 = IX ExF fx-el = l/2X}, respectively. The following relations are well-known consequences of the Jordan identity and are put here for easy reference: