TITLE: Is there a ring isomorphism between $M_n(D)$ and $M_m(D)$ where $n\neq m$ and $D$ is a division ring?
QUESTION [6 upvotes]: Is there a ring isomorphism between $M_n(D)$ and $M_m(D)$ where $n\neq m$ and $D$ is a division ring?

I know that this is impossible if we talk about left $D$-vector space homomorphisms (because of dimension over $D$ must be the same).
What if we think about just a ring isomorphism?

REPLY [4 votes]: (This is related to the same idea Hanno is using, but I like to say it this way.)
$M_n(D)$, as a right module over itself, has a composition series of length $n$, and the same is true for $m$. But the length of the composition series is uniquely defined, and isomorphic rings are going to have identical lattices of right ideals, so $m\neq n$ is not possible.