Example

Theorem 4 applies in the following two specific cases:

a) Let D be a field and Z its center. If D has finite degree over Z, then every automorphism of D that fixes the elements of Z is an inner automorphism. The assumption that D has finite degree over Z is essential (VIII, p. 269, Exercise 4).

b) Let V be a finite-dimensional vector space over the field K. Every automorphism of the K-algebra End$_K(V)$ is an inner automorphism; this result extends to the case when the space V is not finite-dimensional over K (VIII, p. 272, Exercise 13).

In particular, every automorphism of a matrix algebra $\mathbf{M}_n(K)$ (with $n\geqslant$ 1) is an inner automorphism. This result admits the following generalization.