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

(a)  $R$ is isomorphic to $M_{m\times m}(T)$ and $S$ is isomorphic to $M_{n\times n}(T)$ and $m\cdot r = n\cdot s$
<br>or<br>
(b)  $R$ and $S$ both have exactly one element

then $M_{r\times r}(R)$ is isomorphic to $M_{s\times s}(S)$.
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Based on the last comment on Tom Goodwillie's answer to [my earlier question](http://mathoverflow.net/questions/38385/exotic-isomorphism-of-matrix-rings), (a) and (b) apparently do not exhaust the cases in which $M_{r\times r}(R)$ is isomorphic to $M_{s\times s}(S)$.
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What are examples to show that?  Can they be finite?
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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?