Theorem 3 (Skolem–Noether)

Let A and B be simple K-algebras and Z(A) and Z(B) their centers. Suppose that the algebra B has finite degree over K and that the algebra $Z(A)\otimes_KZ(B)$ is a field (which is, in particular, the case when A or B is central). Let $f$ and $g$ be K-algebra homomorphisms from B to A. There exists an inner automorphism $\theta$ of A such that $g=\theta \circ f$.

By Lemma 1 of VIII, p. 254, it suffices to prove that the $(B,A)$-bimodules $A^f$ and $A^g$ are isomorphic. Now, we can view $A^f$ and $A^g$ as left modules over the algebra $C = B\otimes_KA^o$, which is simple by Proposition 7 of VIII, p. 221. As right A-modules, $A^f$ and $A^g$ are isomorphic to $A_d$, hence have finite length because the ring A is simple (VIII, p. 121, Corollary 1). A fortiori, $A^f$ and $A^g$ are C-modules of finite length. Let S be a simple C-module; there exist strictly positive integers $m$ and $n$ such that $A^f$ is isomorphic to $S^m$ and $A^g$ to $S^n$. The right A-module S therefore has nonzero finite length. Since the right A-modules underlying $A^f$ and $A^g$ are isomorphic, they have the same length; we therefore have $m=n$, so that the C-modules $A^f$ and $A^g$ are isomorphic.