A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS

F Soheilnia,Sh Payrovi,A Behtoei

doi:10.24330/ieja.852234

Abstract

Let $R$ be a commutative ring with nonzero identity and let $M$ be a unitary $R$-module. The essential graph of $M$, denoted by $EG(M)$ is a simple undirected graph whose vertex set is $Z(M)\setminus {\rm Ann}_R(M)$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm Ann}_{M}(xy)$ is an essential submodule of $M$. Let $r({\rm Ann}_R(M))\not={\rm Ann}_R(M)$. It is shown that $EG(M)$ is a connected graph with ${\rm diam}(EG(M))\leq 2$. Whenever $M$ is Noetherian, it is shown that $EG(M)$ is a complete graph if and only if either $Z(M)=r({\rm Ann}_R(M))$ or $EG(M)=K_{2}$ and ${\rm diam}(EG(M))= 2$ if and only if there are $x, y\in Z(M)\setminus {\rm Ann}_R(M)$ and $\frak p\in{\rm Ass}_R(M)$ such that $xy\not \in \frak p$. Moreover, it is proved that ${\rm gr}(EG(M))\in \{3, \infty\}$. Furthermore, for a Noetherian module $M$ with $r({\rm Ann}_R(M))={\rm Ann}_R(M)$ it is proved that $|{\rm Ass}_R(M)|=2$ if and only if $EG(M)$ is a complete bipartite graph that is not a star.

Highlights

The concept of the zero-divisor graph of a commutative ring was introduced and studied by I
For a Noetherian R-module M with r(AnnR(M )) = AnnR(M ), we show that EG(M ) is a connected graph with diam(EG(M )) ≤ 2 and gr(EG(M )) ∈ {3, ∞} (Theorem 2.6)
For a Noetherian Rmodule with r(AnnR(M )) = AnnR(M ), we show that Γ(M ) = EG(M ) (Theorem 4.6), where Γ(M ) denotes the zero divisor graph of M

Summary

Introduction

The concept of the zero-divisor graph of a commutative ring was introduced and studied by I. The zero-divisor graph of R, which is the graph with vertex set Z∗(R) = Z(R) \ {0} and two distinct vertices x and y are adjacent if and only if xy = 0, has been studied by many authors (see [1,3,4]). The essential graph of R is a simple undirected graph, denoted by EG(R), with vertex set Z∗(R) and two distinct vertices x and y are adjacent if and only if AnnR(xy) is an essential ideal. We associate a graph to the module M , denoted by EG(M ), with vertex set Z(M ) \ AnnR(M ) and two distinct vertices x, y ∈ Z(M ) \ AnnR(M ) are adjacent if and only if AnnM (xy) is an essential submodule of M.

Properties of the essential graph for modules

Relations between the zero divisor graph and the essential graph