Noncommutative Jordan Rings

Abstract

Heretofore most investigations of noncommutative Jordan algebras have been restricted to algebras over fields of characteristic $\ne 2$ in order to make use of the passage from a noncommutative Jordan algebra $\mathfrak {A}$ to the commutative Jordan algebra ${\mathfrak {A}^ + }$ with multiplication $x \cdot y = \frac {1}{2}(xy + yx)$. We have recently shown that from an arbitrary noncommutative Jordan algebra $\mathfrak {A}$ one can construct a quadratic Jordan algebra ${\mathfrak {A}^ + }$ with a multiplication ${U_x}y = x(xy + yx) - {x^2}y = (xy + yx)x - y{x^2}$, 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 $\mathfrak {A}$ to ${\mathfrak {A}^ + }$ 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 $\ne 2$, 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.