A conception of zero-divisor graph for categories of modules

Abstract

We introduce and study zero-divisor graphs in categories of left modules over a ring R, i.e. R- MOD . The vertices of Γ(R- MOD ) consist of all nonzero morphisms in R- MOD which are not isomorphisms. Two vertices f and g are adjacent if f ◦ g = 0 or g ◦ f = 0. We observe that these graphs are connected and their diameter is equal or less than four. We prove that Γ(R- MOD ) = 3 if and only if R is a right and left perfect ring and R/J(R) is simple artinian. We also characterize all vertices with complements and that when a kernel or a co-kernel can be a complement for a morphism. Some discussions will be made on radius of these graphs, their clique and chromatic numbers.