I am trying to show that if $R \cong M_n(D)$ as rings where $D$ is a division ring, then $R \cong S \oplus … \oplus S$ where $S$ is a simple left $R$-module (isomorphic as modules)
I know that $M_n(D)= C_1 \oplus …\oplus C_n $ where $C_i$ is the ith column with zeroes everywhere else. Each $C_i$ is isomorphic as modules to $D^n$ which is simple as a left $M_n(D) $ module and so each $C_i $ is a simple left ideal in $M_n(D)$.
So letting $\phi :M_n (D) \rightarrow R $ be a ring isomorphism we have $\phi (R)= \phi (C_1 \oplus … \oplus C_n)= \phi (C_1) \oplus … \oplus \phi (C_n) $ and each $\phi (C_i ) $ is a simple left ideal in $R$. But now what I need to show is that each $\phi (C_i )$ is isomorphic to each other as modules and then that would complete the proof but I don’t know how to show this final part.