Let $k$ be a number field. Let $B$ be a central division $k$-algebra.

Let us consider an isomorphism $\varphi: B\otimes_k \mathbb{C}\overset{\sim}{\to} M_n(\mathbb{C})$.

Let $\Lambda$ be a subring of $B$ which is also a free $\mathcal{O}_k$-module. Is $\varphi(\Lambda\otimes 1)\cap U_n(\mathbb{C})$ finite ?

In fact, the question boils down to the following one: let $R$ be a subring of $M_n(\mathbb{C})$ , such that $R=\mathcal{O}_kM_1\oplus\cdots\oplus \mathcal{O}_k M_r$, where $M_1,\ldots,M_r$ are $\mathbb{C}$-linearly independent. Is $R\cap U_n(\mathbb{C})$ finite ?

Unfortunately, $R$ is not necessarily closed and discrete, so if it is true, some other kind of arguments have to be used. I think the answer is "yes", and maybe it is due to the fact that $U_n(\mathbb{C})$ is a compact Lie group, but I would need someone to confirm.

Thanks in advance !