A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS

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