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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 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 (i.e. the ACCP), is an atomic domain. Although the converse is claimed to hold in Cohn's paper,〔P.M. Cohn, Bezout rings and their subrings; Proc. Camb. Phil.Soc. 64 (1968) 251–264〕 this is known to be false.〔A. Grams, Atomic rings and the ascending chain condition for principal ideals. Proc. Cambridge Philos. Soc. 75 (1974), 321–329.〕 The term "atomic" is due to P. M. Cohn, who called an irreducible element of an integral domain an "atom". ==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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「atomic domain」の詳細全文を読む スポンサード リンク
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