Commutative algebra

Atomic domain

In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in at least one way as a finite product of irreducible elements. Atomic domains are 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 element. 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 (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 an irreducible element of an integral domain an "atom". (Wikipedia).

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Related pages

Divisibility (ring theory) | Infinite product | Irreducible element | Prime number | Converse (logic) | Integral domain | Ascending chain condition on principal ideals | Mathematics | Unit (ring theory) | Integer | Fundamental theorem of arithmetic | Unique factorization domain | Noetherian ring | Prime element | Ring (mathematics) | Principal ideal | Ring theory