Opuscula Math. 39, no. 4 (2019), 577-588

Opuscula Mathematica

The intersection graph of annihilator submodules of a module

S.B. Pejman
Sh. Payrovi
S. Babaei

Abstract. Let \(R\) be a commutative ring and \(M\) be a Noetherian \(R\)-module. The intersection graph of annihilator submodules of \(M\), denoted by \(GA(M)\) is an undirected simple graph whose vertices are the classes of elements of \(Z_R(M)\setminus \text{Ann}_R(M)\), for \(a,b \in R\) two distinct classes \([a]\) and \([b]\) are adjacent if and only if \(\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0\). In this paper, we study diameter and girth of \(GA(M)\) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that \(GA(M)\) is complete if and only if \(Z_R(M)\) is an ideal of \(R\). Also, we show that if \(M\) is a finitely generated \(R\)-module with \(r(\text{Ann}_R(M))\neq \text{Ann}_R(M)\) and \(|m-\text{Ass}_R(M)|=1\) and \(GA(M)\) is a star graph, then \(r(\text{Ann}_R(M))\) is not a prime ideal of \(R\) and \(|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1\).

Keywords: prime submodule, annihilator submodule, intersection annihilator graph.

Mathematics Subject Classification: 13C05, 13C99.

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  • Communicated by Adam Paweł Wojda.
  • Received: 2017-09-25.
  • Revised: 2018-10-29.
  • Accepted: 2019-02-26.
  • Published online: 2019-05-23.
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Cite this article as:
S.B. Pejman, Sh. Payrovi, S. Babaei, The intersection graph of annihilator submodules of a module, Opuscula Math. 39, no. 4 (2019), 577-588, https://doi.org/10.7494/OpMath.2019.39.4.577

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