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...
Saved in:
| Main Authors: | , , |
|---|---|
| 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 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| 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 |