Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2024; 61(3): 797-811

Online first article May 21, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230409

Copyright © The Korean Mathematical Society.

A torsion graph determined by equivalence classes of torsion elements and associated prime ideals

Reza Nekooei, Zahra Pourshafiey

Shahid Bahonar University of Kerman; Shahid Bahonar University of Kerman

Abstract

In this paper, we define the torsion graph determined by equivalence classes of torsion elements and denote it by $A_{E}(M)$. The vertex set of $A_{E}(M)$ is the set of equivalence classes $\{[x]\mid x\in T(M)^{*}\}$, where two torsion elements $x, y\in T(M)^{*} $ are equivalent if $ann(x)= ann(y)$. Also, two distinct classes $[x]$ and $[y]$ are adjacent in $A_{E}(M)$, provided that $ann(x)ann(y)M=0$. We shall prove that for every torsion finitely generated module $M$ over a Dedekind domain $R$, a vertex of $A_{E}(M)$ has degree two if and only if it is an associated prime of $M$.

Keywords: Associated prime ideals, Dedekind domain, zero-divisor graph, chromatic number, Clique number

MSC numbers: 13F05, 16D10, 05C15, 05C69