the zero-divisor graph of a module

let $r$ be a commutative ring with identity and $m$ an $r$-module. in this paper, we associate a graph to $m$, say ${gamma}({}_{r}m)$, such that when $m=r$, ${gamma}({}_{r}m)$ coincide with the zero-divisor graph of $r$. many well-known results by d.f. anderson and p.s. livingston have been generalized for ${gamma}({}_{r}m)$. we show that ${gamma}({}_{r}m)$ is connected with ${diam}({gamma}({}_{r}m))leq 3$ and if ${gamma}({}_{r}m)$ contains a cycle, then $gr({gamma}({}_{r}m))leq 4$. we also show that ${gamma}({}_{r}m)=emptyset$ 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, $gr{gamma}({}_{r}m)=infty$ if and only if ${gamma}({}_{r}m)$ is a star graph. finally, we study the zero-divisor graph of free $r$-modules.