Non-commutative Hamilton–Cayley theorem and roots of characteristic polynomials of skew maximal period linear recurrences over Galois rings

Let $p$ be a prime number, $R = \mathrm{GR}(q^d, p^d)$, where $q = p^r$, be a Galois ring, $S = \mathrm{GR}(q^{nd}, p^d)$ be its extension. We prove a non-commutative generalization of the well-known Hamilton–Cayley theorem. Using this result we prove the existence of roots in some extension $\mathcal{K}$ of $\check{S}$ for characteristic polynomials of skew maximal period linear recurrent sequences over $S$. Also for these polynomials we investigate the structure of the set of their roots.