Exercise 20, § 5

Let A be a ring and P, P' two faithful finitely generated projective A-modules. Show that, if 8: $\mathrm{End}_A(P) \to \mathrm{End}_A(P')$ is an A-algebra isomorphism, there exists a projective A-module F of rank 1 and an A-isomorphism

such that $\theta(u) = \phi \circ (u \otimes 1) \circ \phi^{-1}$ 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 8; to prove that $\mathrm{Hom}_B(P, P')$ is a projective A-module of rank 1, use Theorem 2 of no. 3 and Algebra, Chapter VIII, § 5, Exercise 7 (c).)