Half-factorial Domain
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically ring theory, an atomic domain or factorization domain is an
integral domain In mathematics, 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 setting for studying divisibilit ...
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 an integral 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. The irreducible elements are the terminal elements of a factor ...
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 In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish ...
. 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, partially ordered by inclusion. The ascending chain condition on principal ideals (abbrevia ...
(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 In algebra, an irreducible element of an integral 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. The irreducible elements are the terminal elements of a factor ...
of an integral domain an "atom".


Motivation

In this section, a
ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 is prime or can be represented uniquely as a product of prime numbers, ...
. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds. Since a
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 ...
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 (mathematics), 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 ...
s, and second, that this product is unique up to rearrangement (and multiplication by
units Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
). 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
integral domain In mathematics, 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 setting for studying divisibilit ...
. If every non-zero non-unit ''x'' of ''R'' can be written as a product of
irreducible element In algebra, an irreducible element of an integral 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. The irreducible elements are the terminal elements of a factor ...
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 ring theory. 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 factorization). 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 Commutative algebra