Counterexamples for some results in "On the module intersection graph of ideals of rings"

Authors

  • Farideh Heydari Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran https://orcid.org/0000-0002-8944-4729
  • Soheila Khojasteh Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

DOI:

https://doi.org/10.33044/revuma.3826

Abstract

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R)^*$ be the set of all nontrivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I(R)^*$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IM\cap JM\neq 0$. In this note, we provide counterexamples for some results proved in a paper by Asir, Kumar, and Mehdi [Rev. Un. Mat. Argentina 63 (2022), no. 1, 93–107]. Also, we determine the girth of $G_M(R)$ and derive a necessary and sufficient condition for $G_M(R)$ to be weakly triangulated.

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References

T. Asir, A. Kumar, and A. Mehdi, On the module intersection graph of ideals of rings, Rev. Un. Mat. Argentina 63 no. 1 (2022), 93–107.  DOI  MR  Zbl

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F. Heydari, The $M$-intersection graph of ideals of a commutative ring, Discrete Math. Algorithms Appl. 10 no. 3 (2018), article no. 1850038.  DOI  MR  Zbl

S. Khojasteh, The intersection graph of ideals of $ℤ_m$, Discrete Math. Algorithms Appl. 11 no. 4 (2019), article no. 1950037.  DOI  MR  Zbl

R. Nikandish and M. J. Nikmehr, The intersection graph of ideals of $ℤ_n$ is weakly perfect, Util. Math. 101 (2016), 329–336.  MR  Zbl

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Published

2025-05-21

Issue

Vol. 68 No. 1 (2025)

Section

Article

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Copyright (c) 2025 Farideh Heydari, Soheila Khojasteh

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