Do there exist $m,n \in N$ such that the rings of matrices $M_m(\mathbb{R})$ and $M_n(\mathbb{C})$ are isomorphic?

Question

Note: I am looking for insight, and not outright proofs.

My only meaningful observation, here, is that $\mathbb{C} \cong \mathbb{R}^2$ as rings, and that there is an embedding $\mathbb{C} \rightarrow M_n(\mathbb{C})$ and $\mathbb{R} \rightarrow M_m(\mathbb{R})$.

Answer 1

No: isomorphic rings have isomorphic centers, and the center of $M_n(F)$ is isomorphic to $F$.

With the fact that $\mathbb R$ and $\mathbb C$ are nonisomorphic, this gives a complete answer.