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 γ(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 γ(m) is a planar graph, then m ∼= m1 ⊕m2 for a multiplication r-module m with |m ax(m)|̸= 1.