Zero divisor graphs for modules over commutative rings

Abstract

In this article, we give several generalizations of the concept of zero-divisor elements in a commutative ring with identity to modules. Then, for each R-module M, we associate three undirected (simple) graphs Γ∗(RM) ⊆ Γ(RM) ⊆ Γ∗(RM) which, for M = R, all coincide with the zero-divisor graph of R. The main objective of this paper is to study the interplay of module-theoretic properties of M with graph-theoretic properties of these graphs. © 2012 Rocky Mountain Mathematics Consortium.