JOURNAL OF ALGEBRA 18, 103-111 (1971) A Characterization of the Radical of a Jordan Algebra KEVIN MCCRIMMON Defiartnzent of Mathematics, University of Virginia, Charlottesville, Virginia Commzlnicated by Nathan Jacobson Received April 10, 1970 The Jacobson radical of a Jordan algebra has been defined [5] as the maximal ideal consisting entirely of quasi-invertible elements. In this paper we shall obtain a characterization of the radical as the set of properly quasi- invertible elements, in analogy with the case of associative algebras. An element is properly quasi-invertible if it is quasi-invertible in all homotopes. (We show this characterization also works in the associative case). We apply our characterization of the radical of J to describe the radical of U,J, e an idempotent in J, and the radical of an ideal R C J. Throughout we use the notations and terminology of [4] for quadratic Jordan algebras over an arbitrary ring of scalars @. We recall the basic axioms for the composition U,y in the case of unital algebras: Algebras without unit can be defined [7]; an y such algebra J can be imbedded as an ideal in a unital algebra J’ = dsl + J. If % is an associative algebra we obtain a Jordan algebra 2X+ from 9l by taking 7YZ y = X~X. 1. THE ASSOCIATIVE MOTIVATION It is well known that the Jacobson radical of an associative algebra % consists precisely of the properly quasi-immtibZe (p.q.i.) elements, those z for which all ax (equivalently all xa) are quasi-invertible (q-i.) in the sense that 1 - ax is invertible in the algebra 2l’ obtained by adjoining a unit to 2l. It is also known [6] that the Jacobson radical of the Jordan algebra Zlf coincides with that of the associative algebra PI. Now the condition that a~ be q.i. cannot be formulated in Jordan terms since az is not a Jordan product. 103