A GRAPH ASSOCIATED WITH GROUP OF UNITS AND IRREDUCIBLE ELEMENTS IN THE RING OF INTEGER MODULO n

Abstract

A nonzero nonunit $a$ in the ring $\mathbb{Z}_n$ is called an  irreducible element if $a = bc$ implies that either $b$ or $c$ (not both) is a unit in $\mathbb{Z}_n$. We define a graph in which the group of units in $\mathbb{Z}_n$ is a vertex-set and the set of all pairs of units which are both factors of an irreducible element is an edge-set. We study this graph to an extent of determining some of its properties such as the girth, circumference, cliques, cycles and connectivity. Let $n=\prod_{i=1}^{k}p_i^{\alpha_i}$ with some $\alpha_i>1$ and define $\mathcal{P}=\{p_i | p_i \text{ is a prime factor of } n \text{ with } \alpha_i>1\}$. We prove that this graph is $\sum_{p\in\mathcal{P}}(p-1)$-regular and contains isomorphic components, each of which is a union of $K_p$ where $p\in \mathcal{P}$. More importantly, we use this graph to conclude that the distribution of units in factors of irreducible elements of $\mathbb{Z}_n$ is the same if $|\mathcal{P}|=1$ and different if $|\mathcal{P}|>1$.