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On the compressed essential graph of a module over a commutative ring

S. H. Payrovi

https://orcid.org/0000-0003-1186-451X

Department of Mathematics, Imam Khomeini International University, P. O. Box: 34149-1-6818, Qazvin, Iran

E-mail Address: shpayrovi@sci.ikiu.ac.ir

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S. Babaei

Department of Mathematics, Imam Khomeini International University, P. O. Box: 34149-1-6818, Qazvin, Iran

E-mail Address: sakinehbabaei@gmail.com

Corresponding author.

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E. Sengelen Sevim

Eski Silahtaraga Elektrik Santrali, Kazim Karabekir, Istanbul Bilgi University, Cad. No: 2/1334060, Eyup Istanbul, Turkey

E-mail Address: esra.sengelen@bilgi.edu.tr

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https://doi.org/10.1142/S1793557121501254Cited by:0 (Source: Crossref)

Abstract

Let $R$ be a commutative ring and $M$ be an $R$ -module. The compressed essential graph of $M$ , denoted by $E G (M)$ is a simple undirected graph associated to $M$ whose vertices are classes of torsion elements of $M$ and two distinct classes $[m]$ and $[m^{'}]$ are adjacent if and only if ${Ann}_{R} (m) + {Ann}_{R} (m^{'})$ is an essential ideal of $R$ . In this paper, we study diameter and girth of $E G (M)$ and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that $ω (E G (M)) = | {Ass}_{R} (M) |$ , whenever $R$ is Noetherian and $M$ is a finitely generated multiplication module with $r ({Ann}_{R} (M)) = 0$ .

Communicated by H. Tanaka

Keywords:

AMSC: 13Cxx, 05C25

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