If $U\subseteq\mathbf{R}^n$ is an additive subgroup, discrete wrto the induced topology, then $U$ is a finitely generated abelian group.

Is it true that if $U\subseteq\mathbf{A}_K^n$ is an additive subgroup, with $\mathbf{A}_K$ the adéle ring of a number field, $U$ discrete wrto the induced topology, then $U$ is a finitely generated abelian group?

e.g. If $U = \mathcal{O}_K$, the ring of integers of a number field $K$, then $U$ is a finitely generated abelian group, and it's discrete both in $\mathbf{A}_K$ and in $\mathbf{R}\otimes_{\mathbf{Z}}\mathcal{O}_K\simeq\mathbf{R}^{[K:\mathbf{Q}]}$. If $\mathcal{O}_K^{\times}$ is the unit group of $\mathcal{O}_K$, the image $U$ of $\mathcal{O}_K^{\times}$ in the trace-zero hyperplane in $\mathbf{R}^{r_1 + r_2}$, an $r_1+r_2-1$ dimensional $\mathbf{R}$-vector space, is discrete.