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How can we find a monic polynomial with the smallest degree in left ideal of $\mathrm{Mat}(F[x])$?
Let $F$ be a finite field, $R=F[x]$ be a polynomial ring and $K = \mathrm{Mat}_n(R)$ be a full matrix ring over $R$. We identify the ring $K$ with the ring $\mathrm{Mat}_n(F)[x]$, for example
$$
\left(...
