Opuscula Math. 39, no. 4 (2019), 577-588
https://doi.org/10.7494/OpMath.2019.39.4.577

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.
• Revised: 2018-10-29.
• Accepted: 2019-02-26.
• Published online: 2019-05-23.