Why is the nontrivial left ideal of $M_ 2(D)$ minimal?

Question

I'm totally confused by this question. I appreciate any help or answer. thanks in advance why is the nontrivial left ideal of M_2(D) minimal? that D is a division ring

Answer 1

$M_2(D)$ has a composition length of $2$. This is one such composition series:

$$ \begin{bmatrix}0&0\\0&0\end{bmatrix}\subseteq \begin{bmatrix}0&D\\0&D\end{bmatrix}\subseteq M_2(D) $$

It's easy to check that the module in the center is a simple left $M_2(D)$ module. Since you can find a composition series through any nontrivial left ideal, we see that every nontrivial left submodule is both a minimal left and maximal left ideal.