Theorem. (Azumaya – Nakayama, 1947) Let be a central simple -algebra and let be an arbitrary -algebra. If is a two-sided ideal of then
Proof. It is obvious for and so we may assume that Let
Consider the natural -algebra homomorphism
Claim 1. To see this, let Let be a -basis of which contains 1. Let be such that So for some integer and Then
Thus for some we have and for all But then and hence
Claim 2. This is easy to see: since is clearly onto, is a two-sided ideal of Thus, by the theorem we proved in part (3), we must have either which is not true by cliam 1, or
Claim 3. This will be a result of claim 2 if we prove But it is obvious that and so we only need to show that Let Let be a -basis for Then for some positive integer and some Then
and thus, since are linearly independent, for all Hence for all and so
Claim 4. The reason is that if then because and is an ideal of
Corollary. 1) If is a central simple -algebra and is a simple -algebra, then is a simple -algebra.
2) If both and are central simple -algebras, then is a central simple -algebra.
Proof. Part 1) is a trivial result of the above theorem. Part 2) follows from part 1) and Lemma 2.
Example. Let be the division algebra of quaternions over Then
Proof. Let The first part of the above corollary implies that is simple. Moreover, by Lemma 2, and so is a central simple -algebra. Hence, by Remark 2, for some integer Since is a field extension of we have and thus Hence