THE ZERO-DIVISOR GRAPH OF A MODULE

Abstract. Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ??(RM), such that when M = R, ??(RM) coincide with the zero-divisor graph of R. Many well-known results by D. F. Anderson and P. S. Livingston, have been generalized for ??(RM). We will show that ??(RM) is connected with diam(??(RM))  3, and if ??(RM) contains a cycle, then gr(??(RM))  4. We will also show that ??(RM) = ? if and only if M is a prime module. Among other results, it is shown that for a reduced module M satisfying DCC on cyclic submodules, gr (??(RM)) = 1 if and only if ??(RM) is a star graph. Finally, we study the zero-divisor graph of free R- modules.