The M-regular graph of a commutative ring

Year, pages: 2015, 1 - 12

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 $M$-regular graph of $R$, denoted by M$-$\Reg(Γ(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 $M$-$\Reg(Γ(R))$ 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 $M$-$\Reg(Γ(R))$. 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 $M$-$\Reg(Γ(R))$ is finite, then $R$ is finite.

Nikmehr, M., Heydari, F. 2015. The M-regular graph of a commutative ring. In Mathematica Slovaca, vol. 65, no.1, pp. 1-12. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0001

Nikmehr, M., Heydari, F. (2015). The M-regular graph of a commutative ring. Mathematica Slovaca, 65(1), 1-12. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0001