Exercise 7, § 3

Let $p$ be a prime number and let $\mathbf{A}$ denote the ring of rational numbers of the form $r/s$, where $s$ is coprime to $p$ (VII, p. 48, Ex. 4). Let $u = (1, 0)$ and $v = (0, 1)$ be the elements of the canonical basis of the vector space $E = \mathbf{Q}^2$ over $\mathbf{Q}$. Recursively define a sequence of elements $u_n$ of $E$ by $u_0 = u$ and $u_n = p u_{n+1} + v$ for $n \geq 0$. Let F be the A-submodule generated by $v$ and the $u_n$; show that F is not a free module, but contains no nonzero divisible submodule (cf. Ex. 6, a)).