Let K be a field. We consider $K^n$ as a left module of $M_{n, n}(K)$, the ring of matrices of size $n$ over $K$.
1) For any $M_{n, n}(K)$ module homomorphism $ 0 ≠ \phi: M_{n, n}(K) \to K^n$, show that $ker\space \phi$ is always a maximal left ideal.
2) Also, for any maximal left ideal $I \in M_{n,n}(K)$, show that $M_{n,n}(K)/I \cong K^n$, and also show that there's a module homomorphism $\phi$ as described in 1) so that $ker\space\phi = I$.
Thanks in advance. I guess that these maximum ideals are of such shape that they have entries $\in K$ in each line except for just one, in which they only have zeroes? Because these sets of matrices seem to form left-ideals, and also seem to fit the description of maximal ideals, because when also filling the last line with entries $\in K$, they are equal to $M_{n,n}(K)$. But apart from that, I don't really know how to approach these questions.