Properties of co-intersection graph of submodules of a module

Abstract

Let RR be a ring with identity and MM be a unitary left Rmodule. The co-intersection graph of proper submodules of M,Ω(M)M,Ω(M) is an undirected simple graph whose vertices are non-trivial submodule of MM in which two vertices N and K are joined by an edge, if and only if N+K≠MN+K≠M. In this paper, we study several properties of Ω(M)Ω(M). We prove that, if Ω(M)Ω(M) is a path, then Ω(M)≅P2Ω(M)≅P2or Ω(M)≅P3Ω(M)≅P3. We show that, if Ω(M)Ω(M) is a forest, then each component of Ω(M)Ω(M) is complete or star graph. We determine the conditions under which Ω(M)Ω(M) is weakly perfect. Moreover, we introduce the universal vertices and the dominating sets of Ω(M)Ω(M) and their relationship with the non-trivial small submodules of MM.