Set of all kth column matrices in matix ring over a division ring is a minimal left ideal.

Question

Let $D$ be a division ring and $\mathbb{C}_{k}(D)$ be the set of all $k^{th}$ column mtrices in $\mathbb{M}_{m× n}(D)$.Then $\mathbb{C}_{k}(D)$ is a minimal left ideal.

I have shown that it is a left ideal but I am unable to show that it is minimal.

Thanks.

Answer 1

I'm going to write $L$ for your ideal because $\mathbb C_k(D)$ seems too weird.

Hint:

This amounts to showing that for any nonzero $x,y\in L$, there exists a matrix $A\in M_n(D)$ such that $Ax=y$.

What do you think? Given a nonzero $x$ in $D^n$, is there a linear transformation that maps $x$ to $y$?