Revista de la
Unión Matemática Argentina
On the module intersection graph of ideals of rings
Thangaraj Asir, Arun Kumar, and Alveera Mehdi
Volume 63, no. 1 (2022), pp. 93–107

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Let $R$ be a commutative ring and $M$ an $R$-module. The $M$-intersection graph of ideals of $R$ is an undirected simple graph, denoted by $G_{M}(R)$, whose vertices are non-zero proper ideals of $R$ and two distinct vertices are adjacent if and only if $IM\cap JM\neq 0$. In this article, we focus on how certain graph theoretic parameters of $G_M(R)$ depend on the properties of both $R$ and $M$. Specifically, we derive a necessary and sufficient condition for $R$ and $M$ such that the $M$-intersection graph $G_M(R)$ is either connected or complete. Also, we classify all $R$-modules according to the diameter value of $G_M(R)$. Further, we characterize rings $R$ for which $G_M(R)$ is perfect or Hamiltonian or pancyclic or planar. Moreover, we show that the graph $G_{M}(R)$ is weakly perfect and cograph.