Exercise 18, § 5

Let A be a ring, P a faithful finitely generated projective A-module, B the A-algebra $\mathrm{End}_A(P)$, E a left B-module and F the A-module $\mathrm{Hom}_B(P, E)$. Show that the canonical homomorphism

defined in Algebra, Chapter VIII, § 1, no. 4, is bijective. (Reduce it to the case where A is a local ring as in Exercise 17 (c); in this case apply Exercise 9 of Algebra, Chapter VIII, § 1, observing that $F^n$ and $\mathrm{Hom}_B(P, E^n)$ are canonically isomorphic A-modules.)

Show that there exists a strictly increasing bijection of the set of sub-A-modules of F onto the set of sub-B-modules of E (same method).