Proposition 1

Let A be a central simple algebra of finite degree over K. Let B be a K-algebra, and let $f$ and $g$ be algebra homomorphisms from B to A. The following properties are equivalent:

(i) There exists an inner automorphism $\theta$ of A such that $g=\theta \circ f$.

(ii) As left B-modules, $A^f$ and $A^g$ are isomorphic.

By Lemma 1 (VIII, p. 254), property (i) is equivalent to the fact that $A^f$ and $A^g$ are isomorphic as $(B$, A)-bimodules. Since A is finite-dimensional over $K, A^f$ and $A^g$ are B-modules of finite length. Since the center of A is equal to K, the equivalence of (i) and (ii) follows from Lemma 2 of VIII, p. 254 applied to the $(A^o,B^o$)-bimodules $A^f$ and $A^g$.