Automic domain

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 false.