Exercise 8, § 3

Let $p$ be a prime number and $A$ the subring of $\mathbf{Q}$ consisting of the fractions $k/p^n$, where $k \in \mathbf{Z}, n \geq 0$. Show that $A$ is a flat $\mathbf{Z}$-module and that there exists a $\mathbf{Z}$-module $E$ which is not flat but where $A \otimes_{\mathbf{Z}} E$ is a flat $A$-module.