A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS

Year 2021, , 211 - 222, 05.01.2021

F. Soheılnıa Sh. Payrovı A. Behtoeı

https://doi.org/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.

Keywords

Prime submodule, essential submodule, essential graph

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