Perfectness of the essential graph for modules over commutative rings

Let $R$ be a commutative ring and $M$ be an $R$-module. The essential graph of $M$, denoted by $EG(M)$ is a simple graph with vertex set $Z(M) \setminus\operatorname{Ann}(M)$ and two distinct vertices $x,y \in Z(M) \setminus \operatorname{Ann}(M)$ are adjacent if and only if $\operatorname{Ann}_M(xy...

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Bibliographic Details
Main Authors: Fatemeh Soheilnia, Shiroyeh Payrovi, Ali Behtoei
Format: Article
Language:English
Published: Amirkabir University of Technology 2025-02-01
Series:AUT Journal of Mathematics and Computing
Subjects:
Online Access:https://ajmc.aut.ac.ir/article_5327_0630d68c5c10927c46e609e527ddcb78.pdf
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Summary:Let $R$ be a commutative ring and $M$ be an $R$-module. The essential graph of $M$, denoted by $EG(M)$ is a simple graph with vertex set $Z(M) \setminus\operatorname{Ann}(M)$ and two distinct vertices $x,y \in Z(M) \setminus \operatorname{Ann}(M)$ are adjacent if and only if $\operatorname{Ann}_M(xy)$ is an essential submodule of $M$. In this paper, we investigate the dominating set, the clique and the chromatic number and the metric dimension of the essential graph for Noetherian modules. Let $M$ be a Noetherian $R$-module such that ${}|{} {\rm MinAss}_R(M){}|{}=n\geq 2$ and let $EG(M)$ be a connected graph. We prove that $EG(M)$ is a weakly prefect, that is, $\omega(EG(M))=\chi(EG(M))$. Furthermore, it is shown that $\dim (EG(M))= {}|{} Z(M){}|{}-({}|{}\operatorname{Ann}(M){}|{}+2^n)$, whenever $r(\operatorname{Ann}(M) )\not=\operatorname{Ann}(M)$ and $\dim (EG(M))= {}|{} Z(M){}|{}-({}|{}\operatorname{Ann}(M){}|{}+2^n-2)$, whenever $r(\operatorname{Ann}(M) )=\operatorname{Ann}(M)$.
ISSN:2783-2449
2783-2287