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.