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 ?(M), is a simple graph whose vertices are all non-trivial submodules of M and two distinct vertices N1 and N2 are adjacent if and only if N1 ? N2 ?= 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 ?(M) is a planar graph, then M ? = M1 ?M2 for a multiplication R-module M with |Max(M)|?= 1.