The Sum Graph of Non-essential Submodules

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natural numbers n, where Ω(M) is a connected graph, complete graph and has a cyclic. We prove that for a square-free natural number n, Ω(Z_n) is a complete graph. In particular, if n be the product of s distinct prime numbers, then Ω(Z_n) is the complete graph K_s. In addition, we introduce the extended graph Ω_T (M) of Ω(M) for some proper submodule T of M and we investigate about it. Dullay, we define the graph Λ(M) of module M with the set of vertices contain all nontrivial non-small submodules of M. Two distinct vertices N and K are adjacent in Λ(M) if and only if N∩K is a proper non-small submodule of M or N∩K=∘. We prove that, if M be a strongly comultiplication module, then there exists an isomorphism graph Ω(R)≅Λ(M) .