I'm solving exercises of noncommutative ring theory and I have find across the following problem.
If $D$ and $D'$ are division rings and $M_m(D)\simeq M_n(D')$, show that $D\simeq D'$ and $m=n$.
I have already tried to attack in various ways and I understand that this exercise says that, by the Wedderburn-Artin theorem, a simple Artinian ring $R$ is a ring of matrices over a division ring $D$ unique up to isomorphism.
Any suggestion is appreciated. Thank you.