^{}Department of Mathematics, Islamic Azad University, Qazvin Branch, Qazvin, Iran.

Abstract

Let $R$ be a commutative ring with identity and $M$ be an unitary $R$-module. The intersection graph of an $R$-module $M$, denoted by $\Gamma(M)$, is a simple graph whose vertices are all non-trivial submodules of $M$ and two distinct vertices $N_1$ and $N_2$ are adjacent if and only if $N_1\cap N_2\neq 0$. In this article, we investigate the concept of a planar intersection graph and maximal submodules of an $R$-module. In particular, we show that if $\Gamma(M)$ is a planar graph, then $M\cong M_1\oplus M_2$ for a multiplication $R$-module $M$ with $|Max(M)|\neq 1$.