Exercise 20, § 5

Let A be a ring, P, P’ two finite faithful projective a-modules. Now prove that, if $\theta : \mathrm{End}_A(P) \to \mathrm{End}_A(P')$ is an isomorphism of A-algebras, there exists a projective a-module F of rank 1 and an A-isomorphism $\varphi : P' \to P \otimes_A F$, such that $\theta(u) = \varphi^{-1} \circ (u \otimes 1) \circ \varphi$ for all $u \in \mathrm{End}_A(P)$. (Apply Exercise 18 to $E = P'$ considered as an ($\mathrm{End}_A(P)$)-module by means of $\theta$; for proving that $\mathrm{Hom}_B(P, P')$ is a projective a-module of rank 1, use Theorem 2 of No. 3, as well as Alg., Chapter VIII, § 5, Exercise 7 c).)