A Zero-Divisor Graph for Modules With Respect to Their (First) Dual

Abstract

Communicated by S. R. Lopez-PermouthLet M be an R-module. We associate an undirected graph Γ(M ) to M in which nonzero elements x and y of M are adjacent provided that xf (y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈ Hom(M, R). We observe that over a commutative ring R, Γ(M ) is connected and diam(Γ(M )) ≤ 3. Moreover, if Γ(M ) contains a cycle, then gr(Γ(M )) ≤ 4. Furthermore if |Γ(M )| ≥ 1, then Γ(M ) is finite if and only if M is finite. Also if Γ(M ) = ∅, then any nonzero f ∈ Hom(M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Γ(P ) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Γ(M ) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

On Graphs Associated With Modules Over Commutative Rings

“…Ghalandarzadeh and Malakooti Rad in [16] extended the notion of zero-divisor graph to the torsion graph associated with a module M over a ring R, whose vertices are the nonzero torsion elements of M such that two distinct vertices a and b are adjacent if and only if (a : M )(b : M )M = 0. Recent generalizations of zero-divisor graphs to module theory can be found in [9,29].…”

A Generalization of the Zero-Divisor Graph for Modules

“…In recent decades, the zero-divisor graphs of commutative rings (in this paper, called the classic zero-divisor graph) have been extensively studied by many authors and have become a major field of research, see for example [3][4][5][6][7][8][9][10][11][12][13][14][15]. Some authors have also extended the graph of zero-divisors to non-commutative rings, see [18] and [2].…”

“…According to [11], m, n ∈ M are adjacent if and only if (mR : R M )(nR : R M )M = 0, which is a direct generalization of the classic zero-divisor graph. In [8] and [9], the authors have associated two different graphs to an R-module M with respect to its first dual, M * = Hom(M, R). Though they are not necessarily generalizations of the classic zero-divisor graph, there are some deep interrelations between these two graphs and the classic one.…”