ARITHMETICS OF JORDAN ALGEBRAS 21
o-order of $l.
q. e. d.
Note that only closure under the squaring operation was used in the proof of
Lemma 1.
PROPOSITION 1. Let G be a finite dimensional associativ e algebra
over K, ^ = G . An order M of J generates an order M1 of G and if G
contains maximal orders then maximal orders of G and P = G coincide.
PROOF. By Lemma 1, M' is an order of ? = G. If G contains
maximal orders then M1 C E some maximal order of G. But E is an
order of G . Therefore if G contains maximal orders then maximal orders of
G and G coincide.
q. e. d.
It should be noted that the existenc e of maximal orders in G implies the
semisimplicity of G. Also if G is a separable associativ e algebra over K
maximal orders exist ([40], p. 201) and any order is contained in a maximal
order. In particular, if G/K is central simple, G is separable and G is
central simple.
Let (G, *)/K be a finite dimensional associativ e algebra with
involution. Then $ = #(G, *) = {a e G| a* = a} is a subalgebra of G . An
o-order F of G is said to be ^-stabl e if F* C F. Since a** = a, F* C F
implies F = F** C F* and F = F*. A ^-stabl e order F is said to be maximal
*-stabl e if F C E a ^-stabl e order of G implies F = E.
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