We will assume again that is a field. We will denote by the center of a ring The following result is one of many nice applications of the Skolem-Noether theorem (see the lemma in this post!). For the definition of derivations and inner derivations of a -algebra see Definition 2 in this post.
Theorem. Every derivation of a finite dimensional central simple -algebra is inner.
Proof. First note that, since is simple, is simple. We also have Finally Thus is also a finite dimensional central simple -algebra. Now, let be a derivation of Define the map by
for all Obviously is -linear and for all we have
So is a -algebra homomorphism and hence, by the lemma in this post, there exists such that is invertible and for all Let
So gives us
Since holds for all we will get from the last two equations that Since we cannot have because then wouldn’t be invertible, one of or has to be invertible in because is a field. We will assume that is invertible because the argument is similar for It now follows, from the first equation in and the fact that that