Noncommutative Jordan Rings

Abstract

Abstract. Heretofore most investigations of noncommutative Jordan algebras have been restricted to algebras over fields of characteristic ^2 in order to make use of the passage from a noncommutative Jordan algebra 91 to the commutative Jordan algebra 91+ with multiplication x-y = i(xy+yx).We have recently shown that from an arbitrary noncommutative Jordan algebra 91 one can construct a quadratic Jordan algebra 91+ with a multiplication Uxy = x(xy+yx) -x2y = (xy+yx)x-yx2, and that these quadratic Jordan algebras have a theory analogous to that of commutative Jordan algebras.In this paper we make use of this passage from 91 to 91+ to derive a general structure theory for noncommutative Jordan rings. We define a Jacobson radical and show it coincides with the nil radical for rings with descending chain condition on inner ideals; semisimple rings with d.c.c. are shown to be direct sums of simple rings, and the simple rings to be essentially the familiar ones.In addition we obtain results, which seem to be new even in characteristic ^ 2, concerning algebras without finiteness conditions. We show that an arbitrary simple noncommutative Jordan ring containing two nonzero idempotents whose sum is not 1 is either commutative or quasiassociative.1. Axioms and basic results. Throughout this paper 91 will denote a noncommutative Jordan algebra over a unital, commutative, associative ring of scalars i>; we obtain Jordan rings by taking <b = Z, the ring of integers, and the usual linear algebras by taking O to be a field. Unless otherwise stated, <ï> will be completely arbitrary, and we impose no finiteness conditions on 9t.We recall the definition [9, p. 472], [11, p. 141]. A noncommutative Jordan algebra 91 is a space together with a bilinear product xy such that the flexible law hold strictly, i.e. they remain valid in any scalar extension 9tn = 9I <8>« ß. This will automatically be the case if these identities hold in 9Í and <J> is a field with at least three elements (as in most previous investigations), and in general it merely amounts to assuming in addition the linearized version

Spectrum-Finite Ideals of Jordan-Banach Algebras

“…For an associative algebra A the socles of A and A + coincide (see [13]). We take A = B(l 2 ), the Banach algebra of bounded linear operators on the Hilbert space / 2 , and consider the operator T in A defined by…”

Representations of simple noncommutative Jordan superalgebras I

“…Schafer proved that a simple finite-dimensional noncommutative Jordan algebra over a field of characteristic 0 is either a Jordan algebra, or a quasiassociative algebra, or a flexible algebra of degree 2 [32]. Oehmke proved an analog of Schafer's classification for simple flexible algebras with strictly associative powers and of characteristic = 2, 3 [25], McCrimmon classified simple noncommutative Jordan algebras of degree > 2 and characteristic = 2 [17,18], and Smith described such algebras of degree 2 [39]. The case of nodal (degree 1) simple algebras of positive characteristic was considered in the articles of Kokoris [13,14], and the case of infinite-dimensional noncommutative Jordan superalgebras was studied by Shestakov and Skosyrskiy [33,37].…”

The algebraic and geometric classification of nilpotent noncommutative Jordan algebras

“…However, the structure theory of this class is far from being nice. Nevertheless, a certain progress was made in the study of structure theory of noncommutative Jordan algebras and superalgebras (see, for example [5,26,[34][35][36][37]). In this paper, our goal is to obtain a complete algebraic and geometric description of the variety of all 4dimensional nilpotent noncommutative Jordan algebras over the complex field.…”