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Let R be a commutative ring with identity and M be, not necessarily torsion-free, R-module. Unique factorization module (UFM) is introduced via U-decomposition and it is shown that M is a cyclic R-module is necessary but not sufficient condition for M to be a UFM. AMS Subject Classification: 13A05, 13C99, 13E99Key Words: unique factorizatition module, U-decomposition * Let D be an integral domain. It is well known that D is a Unique Factorizatition Domain (UFD) if and only if every nonzero non unit of D is a product of irreducibles, and this factorization into irreducibles is unique up to order and associates. There are several generalizations of this notion UFD to unitary modules over commutative rings [2,3,6,7,8]. In [7] the author defined various type unique factorization modules for torsion-free modules over integral domains. Results in [7] are extended in [6]. In [1] the authors presented a detailed study of factorization in commutative rings with zero divisors, and