Exercise 2, § 5

Show that every $\mathbf{Z}$-module which is Hausdorff with the $p$-adic topology ($p$ a prime number) is ideally Hausdorff, but give an example of a finitely generated $\mathbf{Z}$-module which is Hausdorff but not absolutely Hausdorff with respect to $p$ (use Exercise 1 (a)).