sentences:  Since a unique factorization domain is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered.However, one notices that there are two aspects of the fundamental theorem of the arithmetic; that is, any integer is the finite product of prime numbers, as well as that this product is unique up to rearrangement (and multiplication by units).Therefore, it is also natural to ask under what conditions particular elements of a ring can be "decomposed" without requiring uniqueness.