A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS

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.