A ZERO-DIVISOR GRAPH FOR MODULES WITH RESPECT TO THEIR (FIRST) DUAL

Abstract

Let 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.