ARITHMETICS OF QUADRATIC JORDAN ALGEBRAS

M. L. Racine

I. BASIC CONCEPTS

§1 . Definitions and results concerning quadratic Jordan algebras .

Let $ be a commutative associativ e ring with unit 1. A left $-module

V is said to be unital if lx = x for all x c 1/. From now on $-module will be

understood to mean unital $-module. A unital quadratic Jordan algebra over $

is a triple (#, U, 1) where $ is a left $-module, 1 a distinguished element

of % and U a mapping a - U into End # satisfying the following axioms.

a $

2

(QJ1) U is ^-quadratic, that is U = A U , 7\ e $, a e 9 and

Aa a

U , = U ,, - U - U, is $-bilinear in a and b .

a , b a+b a b

(QJ2) U

1

= 1.

( Q J 3 ) U a U = U b U a U b '

b

(QJ4) If V , is defined by xV = a U , then UUVa, , = V^ U, .

v J ' a , b a , b x, b b b b, a b

(QJ5) (QJ1 - 4) hold for f = $® P , for P any unital commutative

$

associativ e algebra over $ .

For all matters concerning quadratic Jordan algebras for which no specific

reference is given we refer to Jacobson [17]. From now on unital quadratic

Jordan algebra will be abbreviated Jordan algebra.

Let B be a $-submodule of ?. B is an inner ideal of $ if 9 U C B.

B is an outer ideal of # if B U . C E , B is an ideal of 9 if it is both an