On the order of a module

Abstract. Let (R,P) be a Noetherian unique factorization domain (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 ordR(M), be the largest integer n such that I(M)⊆Pn. 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 ordR(M)=2.