Motivation
In this section, a ring can be viewed as merely an abstract set in which one can perform the operations of addition an multiplication; analogous to the integers.
The ring of integers (that is, the set of integers with the natural operations of addition and multiplication) satisfy many important properties. One such property is the fundamental theorem of arithmetic. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds. 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. The concept of an atomic domain addresses this.
Read more about this topic: Atomic Domain
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