Planarity of Intersection Graph of submodules of a Module

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_۱ and N_۲ are adjacent if and only if N_۱\cap N_۲\neq ۰. 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_۱\oplus M_۲ for a multiplication R-module M with |Max(M)|\neq ۱‎.‎