In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain, every non-zero non-unit of which can be written (in at least one way) as a (finite) product of irreducible elements. Atomic domains different from unique factorization domains in that this decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a prime. Important examples of atomic domains include the class of all unique factorization domains, and all Noetherian domains. More generally, any integral domain satisfying the ascending chain condition on principal ideals (i.e. the ACCP), is an atomic domain. Although the converse is claimed to hold in Cohn's paper, this is known to be false.
The term "atomic" is due to P. M. Cohn, who called a irreducible element of an integral domain an "atom".
Read more about Atomic Domain: Motivation, Definition, Special Cases
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