In mathematics, more specifically

ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

, an atomic domain or factorization domain is an

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...

in which every non-zero non-unit can be written in at least one way as a finite product of

irreducible element In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements. Relationship with prime elements Irreducible elements should not be confus ...

s. Atomic domains are different from

unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...

s 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 In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring (mathematics), ring, partially ordered by inclusion (set theory), inclusion. The ascending ch ...

(ACCP) is an atomic domain. Although the

converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...

is claimed to hold in Cohn's paper, this is known to be false. The term "atomic" is due to

P. M. Cohn Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commuta ...

, who called an

of an integral domain an "atom".

Motivation

In this section, a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

can be viewed as merely an abstract set in which one can perform the operations of addition and multiplication; analogous to the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s. 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 In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...

. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds. Since a

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: first, that any integer is the finite product of

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s, and second, that this product is unique up to rearrangement (and multiplication by

units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...

). 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.

Definition

Let ''R'' be an

. If every non-zero non-unit ''x'' of ''R'' can be written as a product of

s, ''R'' is referred to as an atomic domain. (The product is necessarily finite, since

infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...

s are not defined in

. Such a product is allowed to involve the same irreducible element more than once as a factor.) Any such expression is called a factorization of ''x''.

Special cases

In an atomic domain, it is possible that different factorizations of the same element ''x'' have different lengths. It is even possible that among the factorizations of ''x'' there is no bound on the number of irreducible factors. If on the contrary the number of factors is bounded for every non-zero non-unit ''x'', then ''R'' is a bounded factorization domain (BFD); formally this means that for each such ''x'' there exists an integer ''N'' such that if with none of the ''x''_''i'' invertible then ''n'' < ''N''. If such a bound exists, no chain of proper divisors from ''x'' to 1 can exceed this bound in length (since the quotient at every step can be factored, producing a factorization of ''x'' with at least one irreducible factor for each step of the chain), so there cannot be any infinite strictly ascending chain of

principal ideal In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...

s of ''R''. That condition, called the ascending chain condition on principal ideals or ACCP, is strictly weaker than the BFD condition, and strictly stronger than the atomic condition (in other words, even if there exist infinite chains of proper divisors, it can still be that every ''x'' possesses a finite factorizationD. D. Anderson, D. F. Anderson, M. Zafrullah, Factorization in integral domains; J. Pure and Applied Algebra 69 (1990) 1–19). Two independent conditions that are both strictly stronger than the BFD condition are the half-factorial domain condition (HFD: any two factorizations of any given ''x'' have the same length) and the finite factorization domain condition (FFD: any ''x'' has but a finite number of non- associate divisors). Every unique factorization domain obviously satisfies these two conditions, but neither implies unique factorization.

References

{{reflist * P.M. Cohn
Bezout rings and their subrings
1968. Commutative algebra