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 γ(rm), such that when m=r, γ(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 γ(rm). we will show that γ(rm) is connected with diam (γ(rm))≤ 3 and if γ(rm) contains a cycle, then (γ(rm))≤4. we will also show that γ(rm)=ø if and only if m is aprime module. among other results, it is shown that for a reduced module m satisfying dcc on cyclic submodules, gr (γ(rm))=∞ if and only if γ(rm) is a star graph. finally, we study the zero-divisor graph of free rmodules.