The dimension graph for modules over commutative rings

Shiroyeh Payrovi

doi:10.4134/CKMS.c220273

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2023; 38(3): 733-740

Online first article July 17, 2023 Printed July 31, 2023

https://doi.org/10.4134/CKMS.c220273

The dimension graph for modules over commutative rings

Shiroyeh Payrovi

Imam Khomeini International University

Abstract

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Abstract

Let $R$ be a commutative ring and $M$ be an $R$-module. The dimension graph of $M$, denoted by $DG(M)$, is a simple undirected graph whose vertex set is $Z(M)\setminus {\rm Ann}(M)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\dim M/(x, y)M=\min\{\dim M/xM, \dim M/yM\}$. It is shown that $DG(M)$ is a disconnected graph if and only if

(i) ${\rm Ass}(M)=\{\mathfrak p, \mathfrak q\}$, $Z(M)=\mathfrak p\cup \mathfrak q$ and ${\rm Ann}(M)=\mathfrak p\cap \mathfrak q$.

(ii) $\dim M=\dim R/\mathfrak p=\dim R/\mathfrak q$.

(iii) $\dim M/xM=\dim M$ for all $x\in Z(M)\setminus {\rm Ann}(M)$. Furthermore, it is shown that ${\rm diam}(DG(M))\leq 2$ and ${\rm gr}({DG(M)})=3$, whenever $M$ is Noetherian with $|Z(M)\setminus {\rm Ann}(M)| \geq 3$ and $DG(M)$ is a connected graph.

Keywords: Krull dimension, dimension graph

MSC numbers: Primary 13Cxx; Secondary 05C25