New representaions of elements of skew linear recurrent sequences via trace function based on the noncommutative Hamilton – Cayley theorem

Let $p$ be a prime number, $R=\mathrm{GR}(q^d,p^d)$ be a Galois ring of cardinality $q^d$ and characteristic $p^d$, where $q = p^r$, $S=\mathrm{GR}(q^{nd},p^d)$ be its extension of degree $n$ and $\mathrm{End}(_RS)$ be a ring of endomorphisms of the module $_RS$. A sequence $v$ over $S$ satisfying a recursion law
$$ \forall i\in\mathbb{N}_0 \colon v(i+m)= \ \psi_{m-1}(v(i+m-1))+\ldots+\psi_0(v(i)), $$
$\psi_0,\ldots,\psi_{m-1}\in \mathrm{End}(_RS),$ is called skew linear recurrent sequence (LRS) over $S$; the maximal period of such sequence is equal to $(q^{mn}-1)p^{d-1}$. Using the trace function for representations of elements of skew LRS of maximal period we show that such LRS may be linearized if the coefficients in the recursion law are pairwise commuting.