Annihilating submodule graph for modules

Document Type: Research Paper

Author

Department of mathematical Sciences, Yasouj university,Yasouj, 75918-74831, IRAN.

Abstract

Let $R$ be a commutative ring and $M$ an‎ ‎$R$-module‎. ‎In this article‎, ‎we introduce a new generalization of‎ ‎the annihilating-ideal graph of commutative rings to modules‎. ‎The‎ ‎annihilating submodule graph of $M$‎, ‎denoted by $\Bbb G(M)$‎, ‎is an‎ ‎undirected graph with vertex set $\Bbb A^*(M)$ and two distinct‎ ‎elements $N$ and $K$ of $\Bbb A^*(M)$ are adjacent if $N*K=0$‎. ‎In‎ ‎this paper we show that $\Bbb G(M)$ is a connected graph‎, ‎${\rm‎ ‎diam}(\Bbb G(M))\leq 3$‎, ‎and ${\rm gr}(\Bbb G(M))\leq 4$ if $\Bbb‎ ‎G(M)$ contains a cycle‎. ‎Moreover‎, ‎$\Bbb G(M)$ is an empty graph‎ ‎if and only if ${\rm ann}(M)$ is a prime ideal of $R$ and $\Bbb‎ ‎A^*(M)\neq \Bbb S(M)\setminus \{0\}$ if and only if $M$ is a‎ ‎uniform $R$-module‎, ‎${\rm ann}(M)$ is a semi-prime ideal of $R$‎ ‎and $\Bbb A^*(M)\neq \Bbb S(M)\setminus \{0\}$‎. ‎Furthermore‎, ‎$R$‎ ‎is a field if and only if $\Bbb G(M)$ is a complete graph‎, ‎for‎ ‎every $M\in R-{\rm Mod}$‎. ‎If $R$ is a domain‎, ‎for every divisible‎ ‎module $M\in R-{\rm Mod}$‎, ‎$\Bbb G(M)$ is a complete graph with‎ ‎$\Bbb A^*(M)=\Bbb S(M)\setminus \{0\}$‎. ‎Among other things‎, ‎the‎ ‎properties of a reduced $R$-module $M$ are investigated when‎ ‎$\Bbb G(M)$ is a bipartite graph‎.

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