Domains

In mathematics, more specifically ring theory, an atomic domain is an integral domain, every non-zero non-unit of which, is a finite product of irreducible elements. Atomic domains are different from unique factorization domains, because this finite decomposition of an element into irreducibles need not be unique; equivalently, not every irreducible element is a prime. Important examples of atomic domains include the class of all unique factorization domains, and all Noetherian domains. In particular, any integral domain satisfying the ascending chain condition on principal ideals (i.e. the ACCP), is a atomic domain. Despite the claim that appears in Cohn's famous paper, the converse is known to be falseThe 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.