Exercise 2, § 1

Let G be a discrete subgroup of $\mathbf{R}^n$. If G is the direct sum of two subgroups H and K, then the intersection of the vector subspaces generated by H and K consists only of 0, and the rank of G is therefore equal to the sum of the ranks of H and K (observe that, for every discrete subgroup G of $\mathbf{R}^n$, the vector subspace of $\mathbf{R}^n$ generated by G is canonically isomorphic to $G \otimes_{\mathbf{Z}} \mathbf{R}$).