ARITHMETICS OF JORDAN ALGEBRAS 3

z ^ 0 is an absolute zero divisor if U = 0; x

G

# is qua si-invertible if

1 - x is invertible. McCrimmon [28] has defined the Jacobson radical ft of

P to be the maximal quasi-invertible ideal of $ (a subset B of J is

qua si-invertible if every element of B is quasi-invertible). ^ is semisimple

if its radical ${$) is zero. McCrimmon has shown that if ft(z) is zero then

$ contains no absolute zero-divisors . So if ^ is finite dimensional over a

field, we know from structure theory that ^ is a direct sum of a finite number

p

of simple summands. A semisimple algebra ^ is separable if $ is semi -

simple for any commutative associativ e extension P of $.

Following McCrimmon [3 0] we define the centroid r ( ^ / $ ) of a Jordan

algebra over $ to be the set of all $-endomorphisms T of ^ such that

(1) TU - U T , U ^ = T 2 U .

x x xT x

The centroid of a simple algebra is a field. We sa y that y is central over

$ if r = $.

Define V = V_ , {xyz} = yU = yU = {zyx} and x o y = 1U =

x l , x ' x, z z , x ' J x x, y

1U = y o x. Letting b = 1, a = y in (QJ4) we obtain xU V = xV U

or yU . = 1U

x, 1 x, y

(2) x o y = l U = 1U = xU = yU = xV = yV .

x, y y, x l , y l , x y x

We recall the following identities

(3) {xyz} + {yxz} = (x o y) o z ([17], p. 1.20)