Let $R$ be a finite ring and $r \in R$ such that $f \in R[X]$ is a min. poly of $r$ over $R$. Ie. $f(r) = 0$ and $f$ has minimal degree. In particular if $g(r) = 0$ and $g \in R[X]$, we have that $f \vert \ g(X)$ in $R[X]$.
Since $R$ is finite, the iteration sequence $f^{1}(r), f^{2}(r), \dots$ eventually repeats at $f^m(r) = f^n(r)$, $m \gt n$. This means that $f \vert \ f^m - f^n$ in $R[X]$.
But we also then have, relatedly, that $X \vert \ (f^{m-1} - f^{n-1})$, $f^{s+1} \vert \ (f^{m+s} - f^{n+s})$, and finally since $(f^{m-s}(X) - f^{n-s}(X))\circ f^s(\alpha) = 0$, that $(X - f^s(\alpha)) \vert \ (f^{m-s}(X) - f^{n-s}(X)), \ \forall s = 0\dots n $.
Can we do anything useful with this information?