Corollary 2

Let $r$ and $r'$ be strictly positive integers, and let D and $D'$ be fields. The rings $\mathbf{M}_r(D)$ and $\mathbf{M}_{r'}(D')$ are isomorphic if and only if we have $r=r'$ and the fields D and $D'$ are isomorphic.

The condition is obviously sufficient.

Conversely, suppose that the rings $B =\mathbf{M}_r(D)$ and $B'=\mathbf{M}_{r'}(D')$ are isomorphic. Since $r$ is the length of $B_s$ and $r'$ that of $B'_s$ (Lemma 2), we have $r=r'$. Moreover, B is Morita equivalent to D and $B'$ to $D'$ (VIII, p. 102, Example 1). Consequently, the fields D and $D'$ are Morita equivalent, hence isomorphic (VIII, p. 111, Proposition 13, c)).