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

Abstract

We know that $\mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $\mathbb{Z}^m_n= \mathbb{Z}_n \times\mathbb{Z}_n\times...\times\mathbb{Z}_n$ be the Cartesian product of $m$ rings $\mathbb{Z}_n$. In this note, we present the action of $SL(m, \mathbb{Z}_n)=\{A \in \mathbb{Z}^{m,m}_{n} : det A \equiv 1 (mod\simn)\}$, where $SL(m, \mathbb{Z}_n)$ for $n\geq 2$ is a group under matrix multiplication modulo $n$, on the ring $\mathbb{Z}^m_n$ as a right multiplication of a row vector of $\mathbb{Z}^m_n$ by a matrix of $SL(m, \mathbb{Z}_n)$ to determine the orbits of the ring $\mathbb{Z}^m_n$. This work is an extension of [1]