We will assume again that k is a field. We will denote by Z(A) the center of a ring A. 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 k-algebra see Definition 2 in this post.

Theorem. Every derivation of a finite dimensional central simple k-algebra A is inner.

Proof. First note that, since A is simple, M_2(A) is simple. We also have Z(M_2(A)) \cong Z(A) = k. Finally \dim_k M_2(A) = 4 \dim_k A < \infty. Thus M_2(A) is also a finite dimensional central simple k-algebra. Now, let \delta be a derivation of A. Define the map f: A \longrightarrow M_2(A) by

f(a) = \begin{pmatrix} a & \delta(a) \\ 0 & a \end{pmatrix},

for all a \in A. Obviously f is k-linear and for all a,a' \in A we have

f(a)f(a') = \begin{pmatrix} aa' & \delta(a)a'+a \delta(a') \\ 0 & aa' \end{pmatrix} = \begin{pmatrix} aa' & \delta(aa') \\ 0 & aa' \end{pmatrix} = f(aa').

So f is a k-algebra homomorphism and hence, by the lemma in this post, there exists v \in M_2(A) such that v is invertible and f(a)=vav^{-1}, for all a \in A. Let

v = \begin{pmatrix} x & y \\ z & t \end{pmatrix}.

So f(a)v=va gives us

\begin{cases} ax+\delta(a)z=xa \\ ay+\delta(a)t=ya \\ az=za \\ at=ta. \end{cases} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*)

Since (*) holds for all a \in A, we will get from the last two equations that z,t \in Z(A)=k. Since we cannot have z=t=0, because then v wouldn’t be invertible, one of z or t has to be invertible in k because k is a field. We will assume that z is invertible because the argument is similar for t. It now  follows, from the first equation in (*) and the fact that z^{-1} \in Z(A), that

\delta(a)=(xz^{-1})a - a(xz^{-1}). \ \Box

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Comments
  1. Mr. A says:

    So finite dimensional simple k-algebras are separable 🙂

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