The M-Regular Graph of a Commutative Ring

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 $M\mathit{\text{-regular}}\mathit{\text{graph}}$ of $R$, denoted by $M$-$Reg(\mathrm{\Gamma }(R))$. 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 $M$-$Reg(\mathrm{\Gamma }(R))$ 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 $M$-$Reg(\mathrm{\Gamma }(R))$. Among other results, we prove that for every Noetherian ring $R$ and every finitely generated module $M$ over $R$, if $2\mathrm{\notin }Z(M)$ and the independence number of the $M$-$Reg(\mathrm{\Gamma }(R))$ is finite, then $R$ is finite.