Theorem 2

Let B be a semisimple ring and Z its center; suppose that B is a finitely generated Z-module. Let A be a right Artinian ring, and let $f$ and $g$ be ring homomorphisms from B to A; write $f_Z$ and $g_Z$ for the restrictions of $f$ and $g$ to Z. The following properties are equivalent:

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

(ii) There exists an inner automorphism $\theta$ of A such that $g_Z=\theta \circ f_Z$.

Since the ring A is right Artinian, $A_d$ is a right A-module of finite length (VIII, p. 6, Theorem 1). Hence $A^f$ and $A^g$ are $(B$, A)-bimodules of finite length. By Lemma 1 (VIII, p. 254), assertion (i) means that $A^f$ and $A^g$ are isomorphic $(B$, A)-bimodules and assertion (ii) that they are isomorphic $(Z,A)$-bimodules. The equivalence of (i) and (ii) therefore follows from Lemma 2 (VIII, p. 254).