Any matrix ring $R_k$ is isomorphic to a subring of the ring $R_{2^k}$

Question

I am reading the paper: "Logical connections between some open problems concerning nil rings" by Jan Krempa. I have a problem trying to understand the last part of Theorem 2. It says: "Any matrix ring $R_k$ is isomorphic to a subring of the ring $R_{2^k}$".

I don't see why is that true and he is not giving any clue of that. Any help is much appreciated.

Answer 1

Why don't you just map things in $R_k$ to matrices in $R_{2^k}$ which have the matrix from $R_k$ in the upper left hand block? You seem to be suggesting it’s ok that it does not share identity.

That's a subring isomorphic to $R_k$.