Theorem 4

Tag 00GW, permanent. Cite it as [Bourbaki, Tag 00GW].

Algebra, VIII, p. 257, § 14. Central Simple Algebras

Let A be a semisimple ring, Z its center, and $u$ an automorphism of A. Suppose that A is a finitely generated Z-module and that we have $u(z) =z$ for every $z$ in Z. Then $u$ is an inner automorphism.

This follows from Theorem 2 of VIII, p. 256 applied with $f=$ Id $_A$ and $g=u$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Exercise 15, § 14
Example, § 14

Cites

Theorem 2 of VIII, p. 256