Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other.

Question

Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other.

I need some hints to prove it.

Thank you very much.

Answer 1

One way to approach the problem: If $M$ is a simple $R$-module, then $M \cong R/I$ for some maximal left ideal $I ⊂ R$. What are the maximal left ideals in $R$?

Elaborating: Show that a maximal left ideal $I ⊂ R $ is already generated by any matrix of maximal rank in $I$ and find that maximal rank. Hint: Use row reduction. Then show that any two matrices with this rank differ¹ by an $R$-module automorphism $R → R$. Hint: Use column reduction.

What can you conclude about any two maximal left ideals in $R$? You can use the first isomorphism theorem for the final step.

¹: They might also differ by some unit of $R$ as well.