The M-Regular Graph of a Commutative Ring

TL;DR

This paper introduces the $M$-regular graph of a commutative ring, generalizing the regular graph concept, and explores its properties, structure, and implications for Noetherian rings and finitely generated modules.

Contribution

It defines the $M$-regular graph of a ring, studies its properties, and establishes conditions linking graph invariants to the finiteness of the ring.

Findings

Determined the girth of the $M$-regular graph.

Provided lower bounds for independence and clique numbers.

Proved that certain conditions imply the ring is finite.

Abstract

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $Z(M)$ be the set of all zero-divisors on $M$. In 2008, D.F. Anderson and A. Badawi introduced the regular graph of $R$. In this paper, we generalize the regular graph of $R$ to the \textit{$M$-regular graph} of $R$, denoted by $M$-$Reg(\Gamma(R))$. It is the undirected graph with all $M$-regular elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(M)$. The basic properties and possible structures of the $M$-$Reg(\Gamma(R))$ are studied. We determine the girth of the $M$-regular graph of $R$. Also, we provide some lower bounds for the independence number and the clique number of the $M$-$Reg(\Gamma(R))$. Among other results, we prove that for every Noetherian ring $R$ and every finitely generated module $M$ over $R$, if $2\notin Z(M)$ and the independence number of the $M$-$Reg(\Gamma(R))$ is finite, then $R$ is finite.