On the order of a module

Abstract

Abstract. Let $(R,P)$ be a Noetherian unique factorization domain (UFD) and M be a finitely generated R-module. Let I(M)<br />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 $ord_R(M) = 2.$