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A torsion-free module M over integral domain R is called unique factorization module (UFM) if the following conditions are satisfied:(1) Every nonzero element x in M has an irreducible factorization, that is, x=a_1a_2...a_nm, with a_1,a_2,...,a_n are irreducible in R and m is irreducible in M, and(2) if x=a_1a_2...a_nm=b_1b_2...b_km' are two irreducible factorizations of x, then n=k, m~m' in M, and we can rearrange the order of the b_i's so that a_i~b_i in R for every i in {1,2,...,n}.The definition of UFM is a generalization of the concept of factorization on the ring which is applied to the module. In this study, we will discuss another definition that is a weakness of UFM, namely the Weakly Unique Factorization Module (w-UFM). First of all, some concepts that play an important role in defining w-UFM are given. After that, the definition and characterization of w-UFM is also given. The results of this study will provide the sufficient and necessary conditions of the w-UFM.