Let $R=M_n(\mathbb F_q )$ be the ring of $n\times n$ matrices over the finite field $\mathbb F_q$. I want to show that every matrix of rank $n-1$ in any maximal left ideal of $R$ generates that maximal left ideal.

I know that $R$ is semi-simple ring. I deduce that every ideal of $R$ was generated by idempotent element, but I don't know how do I use this fact.

Any suggestion?


Finally, I found the solution of my question in Jacobson's book (page 232 of Lectures in Abstract Algebra). However Jacobson used old literature and I restate with modern literature.

Let $D$ be a division ring and n be a natural number. If $H_r$, $0 \leq r \leq n$, denotes the left ideal of $M_n(D)$ containing all matrices whose $d_{ij} = 0$, for every $r \leq j \leq n$, then every left ideal of $M_n(D)$ is similar to $H_r$, for some $r$, $0 \leq r \leq n$ (there exists an invertible matrix $P$, so that every matrix in the left ideal of $M_n(D)$ is of the form $PKP^{−1}$, where $K\in H_r$).


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