Atomic domain
Encyclopedia
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 element
s. Atomic domains different from unique factorization domain
s in that this decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a prime
. 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, this is known to be false.
The term "atomic" is due to P. M. Cohn, who called a irreducible element
of an integral domain an "atom".
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 number
s, 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.
x of R can be written as a product of irreducible element
s, R is referred to as an atomic domain. (The product is necessarily finite, since infinite products 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.
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 ideals of R. That condition, called 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, an atomic domain or factorization domain is an integral domain, every non-zero non-unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
of which can be written (in at least one way) as a (finite) product of irreducible element
Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...
s. Atomic domains different from unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
s in that this decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a prime
Prime element
In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...
. Important examples of atomic domains include the class of all unique factorization domains, and all Noetherian domains
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
. More generally, any integral domain satisfying the ascending chain condition on principal ideals
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...
(i.e. the 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 a irreducible element
Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...
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
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique 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
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
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 number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, as well as that this product is unique up to rearrangement (and multiplication by units
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
). 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. If every non-zero non-unitUnit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
x of R can be written as a product of irreducible element
Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...
s, R is referred to as an atomic domain. (The product is necessarily finite, since infinite products are not defined in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
. 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 nonzero nonunit x, then R is a bounded factorization domain (BFD); formally this means that for each such x there exists an integer N such that with none of the xi invertible implies .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 ideals of R. That condition, called 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.