The Total Graph of a Module Over a Commutative Ring With Respect to Proper Submodules

Abstract

Let R be a commutative ring and M be an R-module with a proper submodule N. The total graph of M with respect to N, denoted by T(ΓN(M)), is investigated. The vertex set of this graph is M and for all x, y belonging to M, x is adjacent to y if and only if x + y ∈ M(N), where M(N) = {m ∈ M : rm ∈ N for some r ∈ R - (N : M)}. In this paper, in addition to studying some algebraic properties of M(N), we investigate some graph theoretic properties of two important subgraphs of T(ΓN(M)) in the cases depending on whether or not M(N) is a submodule of M.

An extension of total graph over a module

“…For a proper submodule [1], the authors introduce a generalization of total graph of a module with respect to the set M.L/. For ½, N is more general than M.L/ since there are R-modules M and ½such that N is not of the form M.L/ for all submodules L of M .…”

“…In section 4, we study the case when N is a (prime) submodule of M . In the final section, we do the case when N is not a submodule of M , and we improve Theorem 3.3 in [7] and Theorem 4.3 in [1] by Theorem 14, and Theorem 3.12 in [5] by Theorem 15.…”

“…In this section, we consider the question of when for an R-module M , the set M.L/ is a union of prime submodules and determine a family of non-primeless R-modules. We refer the reader to [1] for some properties concerning M.L/. Throughout this section, L is a proper submodule of M over the commutative ring…”

A generalization of total graphs of modules

“…x + y ∈ Z(R), where Z(R) denotes the set of zero-divisors of R. The concept of total graphs is a great concept that is usually used in commutative algebra to obtain many interesting graphs in this field. In [1] and [2], A. Abbasi and S. Habibi, gave a generalization of the total graph. They studied in [2] Let G be a simple graph.…”

“…In [1] and [2], A. Abbasi and S. Habibi, gave a generalization of the total graph. They studied in [2] Let G be a simple graph. If there is a path from any vertex to any other vertex of graph G, then G is said to be connected, and G is said to be totally disconnected This work has been supported by the University of Guilan.…”