Corollary

Let A and B be algebras over the field K. Suppose that B is central simple and of finite degree and that A is right Artinian. Let $f$ and $g$ be K-algebra homomorphisms from B to A. There exists an inner automorphism $\theta$ of A such that $g=\theta \circ f$.

In the notation of Theorem 2, we have Z = K and therefore $f_Z= 1_Z=g_Z$.