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?