A note on the action of $SL(m,{\mathbb{Z}}_n)$ on the ring ${\mathbb{Z}}_n^m$

We know that ${\mathbb{Z}}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $$\mathbb{Z}$^m_n= $\mathbb{Z}$_n \times$\mathbb{Z}$_n\times…\times$\mathbb{Z}$_n$betheCartesianproductof$m$rings$$\mathbb{Z}$_n$\mathrm{.}Inthisnote,wepresenttheactionof$SL(m, $\mathbb{Z}$n)={A \in $\mathbb{Z}$^{m,m}{n} : det A \equiv 1 (mod\simn)}$, where $SL(m,{\mathbb{Z}}_n)$ for $n\ge 2$ is a group under matrix multiplication modulo $n$, on the ring ${\mathbb{Z}}_n^m$ as a right multiplication of a row vector of ${\mathbb{Z}}_n^m$ by a matrix of $SL(m, $\mathbb{Z}$_n)$todeterminetheorbitsofthering$$\mathbb{Z}$^m_n$. This work is an extension of [1]