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
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