Exercise 11, § 5

Let P be a finitely generated projective A-module; denote its trace ideal by $\tau (P)$. a) Prove that we have $\tau (P)^2=\tau (P)$ and $\tau (P)P = P$.

b) Prove that the canonical mapping $\tau (P)\otimes_AP\rightarrow P$ is an isomorphism.