Abstract. Let (R,P) be a Noetherian unique factorization do-main (UFD) and M be a finitely generated R-module. Let I(M)be the first nonzero Fitting ideal of M and the order of M, denotedord_R(M), be the largest integer n such that I(M) ⊆ P^n. In thispaper, we show that if M is a module of order one, then either Mis isomorphic with direct sum of a free module and a cyclic moduleor M is isomorphi...

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}({}_...