Let $R,S,T$ be (associative, with a 1) rings, and $m,n,r,s$ be non-negative integers.

If

(a) $R$ is isomorphic to $M_{m\times m}(T)$ and $S$ is isomorphic to $M_{n\times n}(T)$ and $m\cdot r = n\cdot s$

or

(b) $R$ and $S$ both have exactly one element

then $M_{r\times r}(R)$ is isomorphic to $M_{s\times s}(S)$.

Based on the last comment on Tom Goodwillie's answer to my earlier question, (a) and (b) apparently do not exhaust the cases in which $M_{r\times r}(R)$ is isomorphic to $M_{s\times s}(S)$.

What are examples to show that? Can they be finite?

Is there a relatively simple exhaustion of when $M_{r\times r}(R)$ is isomorphic to $M_{s\times s}(S)$ in terms of $R,S,r,s$? What if $R$ and $S$ are assumed to be finite?