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)
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