We introduce the notion of classical primary submodules that generalizes the concept of primary ideals of commutative rings to modules. Existence and uniqueness of classical primary decompositions in finitely generated modules over one-dimensional Noetherian domains are proved.

Abstract. Let R be a commutative ring with identity and M an Rmodule. In this paper, we associate a graph to M , say Γ(M ), such that when M = R, Γ(M ) is exactly the classic zero-divisor graph.

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.

In this paper, we will extend the notion of zero-divisor graph of commutative rings to zero-divisor graph of abelian groups and study this zero-divisor graph. We characterize the zero-divisor graph of almost all abelian groups. Just a few classes of reduced abelian groups remain untouched in this paper.

The zero-divisor graphs of modules introduced and studied in [S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc. 51(1) (2014) 87–98]. Basic results for zero-divisor graphs of [Formula: see text]-modules were obtained in [M. Baziar, E. Momtahan and S. Safaeeyan, Zero-divisor graph of abelian groups, J. Algebra Appl. 13(6) (2014) 13]. In this paper, zero-divisor graphs in the title are studied. Here, among other things, we generalize results stated in [M. Baziar, E. Momtahan and S. Safaeeyan, Zero-divisor graph of abelian groups, J. Algebra Appl. 13(6) (2014) 13]. Some results for modules over non-integral domains are also obtained.