On the order of a module

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, denoted ord_R(M), be the largest integer n such that I(M) ⊆ P^n. In this paper, we show that if M is a module of order one, then either M is isomorphic with direct sum of a free module and a cyclic module or M is isomorphic with a special module represented in the text. We also assert some properties of M while ord_R(M) = 2.