Throughout this post is a field. Recall that the center of a simple ring is a field (see Remark 1 in here).
Definition. A ring is called a central simple -algebra if is simple and If the dimension of as a vector space over is finite, then is called a finite dimensional central simple -algebra.
Remark 1. Let be a finite dimensional central simple -algebra and let be a descending chain of left ideals of Each is clearly a -vector subspace of and therefore Thus, since the chain must stop at some point. This shows that every finite dimensional central simple -algebra is (left) Artinian. Hence is simple and Artinian and thus, by the Wedderburn-Artin theorem, for some positive integer and some division ring Note that
Let Then, since and
Note. Since we may assume that is the center of and write instead of
Remark 2. If is algebraically closed and is a finite dimensional central simple -algebra, then for some To see this, we first apply Remark 1 to get a division ring with and such that So is algebraically closed. Now if then is an algebraic field extension of because Thus because is algebraically closed. So and hence
Lemma. Let and be two -algebras and let Then
1) if and only if
2) If is central simple, then the centralizer of in is
Proof. 1) If then by Lemma 2
Conversely, if then
and hence Therefore
2) Just to avoid any possible confusion, I mention that we have identified and with and here. It is obvious that every element of commutes with every element of So the centralizer of in contains Conversely, let be in the centralizer of in Let be a -basis for Then there exist such that Since is in the centralizer of we must have for all
Thus and so by Lemma 1, for all Hence and so