The second coordinate sequence of the MP-LRS over nontrivial Galois ring of an odd characteristic

A Galois ring is said to be nontrivial if it is neither a field, nor a residue ring. Divisors and multiples of the minimal polynomial of the second $p$-adic coordinate sequence of a linear recurrent sequence of maximum period (MP LRS) over a nontrivial Galois ring of an odd characteristic are described, as well as their relation to the initial vector of the LRS. As a corollary, nontrivial upper and lower bounds for the rank of the second coordinate sequence of a MP LRS over a nontrivial Galois ring of an odd characteristic are obtained.