Exercise 1, § 6

Let $A,B$ be rings, P an $(A$, B)-bimodule, and Q a $(B$, A)-bimodule. Let $\varphi :$ $P\otimes_BQ\rightarrow A$ be a homomorphism of $(A$, A)-bimodules and $\psi : Q\otimes_AP\rightarrow B$ a homomorphism of $(B$, B)-bimodules satisfying $\varphi (x\otimes u)y=x\psi (u\otimes y)$ and $u\varphi (x\otimes v) =$ $\psi (u\otimes x)v$ for $x, y$ in P and $u, v$ in Q. Suppose that $\varphi$ is surjective.

a) Prove that the A-module P is generating.

b) Prove that $\varphi$ is an isomorphism.

c) Construct an isomorphism of $(B$, A)-bimodules between Q and Hom$_B(P,B)$.