# Why is $M_n(k)$ a semisimple $M_n(k)$-module?

I found in my notes that, for $$k$$ a field,

$$M_n(k)$$ is a direct sum of its column left ideals, so it is a semisimple $$M_n(k)$$-module.

I don't understand the first clause, though I feel it must be some obvious linear algebra.

I mean, what are the column left ideals of $$M_n(k)$$?

I think of the spaces generated by each element of the canonical basis for $$M_n(k)$$, i.e. The matrices with a 1-entry and all the other entries zero, but they're not column ideals.

Let's do $$n=2$$. The column left ideals in this case are $$I_1=\left\{\pmatrix{*&0\\*&0}\right\}$$ and $$I_2=\left\{\pmatrix{0&*\\0&*}\right\}$$ in what I hope is an obvious notation. Then $$M_2(k)=I_1\oplus I_2$$ and moreover $$I_1=M_2(k)A$$ for any nonzero $$A\in I_1$$ (and similarly for $$I_2$$) so that $$I_1$$ is a simple module etc.
• Oh thanks, now that it is written out it's clearer. And that's also a nice way to see the simplicity of $I_1$. Sep 15, 2019 at 18:17