Sufficient and necessary conditions of a module to be the unique factorization modules

An integral domain D is called a unique factorization domain (UFD) if the following conditions are satisfied: (1) Every element of D that is neither O nor a unit can be factored into a product of a finite number of irreducibles, and (2) if p₁, … , p_r and q₁ …, q_s are two factorizations of the same element of D into irreducibles, then r = s and q_j can be renumbered so that p_i and q_i are associates. It has been known the following equivalent conditions: Let D be an integral domain, the following are equivalent: (i) D is a UFD, (ii) D is a GCD domain satisfying the ascending chain condition on principal ideals, and (iii) D satisfies the ascending chain condition on principal ideals and every irreducible element of D is a prime element of D. The concept and the equivalent conditions on UFD, motivate some studies that may apply the concept of factorization to modules in order to obtain a definition of a unique factorization module (UFM). First, the concept of irreducible elements is given in the module which will play an important role in defining the UFM. The definition of primitive elements, pure submodules, least common multiple and greatest common divisor in a module is also given. Next, the definition and characterization of a UFM will be presented. The results of this study is providing the sufficient and necessary conditions of a module to be UFM.