Exercise 18, § 5

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

β:P⊗ A F→E

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

Now prove that there exists a strictly increasing bijective mapping from the set of sub-a-modules of F onto the set of sub-b-modules of E (same method).