In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of
Bourbaki) is 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 ...
in which a statement analogous to 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 ...
holds. Specifically, a UFD 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 set ...
(a
nontrivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
in which the product of any two non-zero elements is non-zero) in which every non-zero non-
unit
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'' (alb ...
element can be written as a product of
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 pri ...
s (or
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), uniquely up to order and units.
Important examples of UFDs are the integers and
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
s in one or more variables with coefficients coming from the integers or from a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
Unique factorization domains appear in the following chain of
class inclusions:
Definition
Formally, a unique factorization domain is defined to be 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 set ...
''R'' in which every non-zero element ''x'' of ''R'' can be written as a product (an
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
if ''x'' is a unit) 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 ''p''
i of ''R'' and a
unit
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'' (alb ...
''u'':
:''x'' = ''u'' ''p''
1 ''p''
2 ⋅⋅⋅ ''p''
''n'' with ''n'' ≥ 0
and this representation is unique in the following sense:
If ''q''
1, ..., ''q''
''m'' are irreducible elements of ''R'' and ''w'' is a unit such that
:''x'' = ''w'' ''q''
1 ''q''
2 ⋅⋅⋅ ''q''
''m'' with ''m'' ≥ 0,
then ''m'' = ''n'', and there exists a
bijective map
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
''φ'' : → such that ''p''
''i'' is
associated Associated may refer to:
*Associated, former name of Avon, Contra Costa County, California
* Associated Hebrew Schools of Toronto, a school in Canada
*Associated Newspapers, former name of DMG Media, a British publishing company
See also
*Associati ...
to ''q''
''φ''(''i'') for ''i'' ∈ .
The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful:
:A unique factorization domain is an integral domain ''R'' in which every non-zero element can be written as a product of a unit and
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 pri ...
s of ''R''.
Examples
Most rings familiar from elementary mathematics are UFDs:
* All
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
s, hence all
Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. ...
s, are UFDs. In particular, the
integers
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 language o ...
(also see
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 ...
), the
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s and the
Eisenstein integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
:z = a + b\omega ,
where and are integers and
:\omega = \f ...
s are UFDs.
* If ''R'' is a UFD, then so is ''R''
'X'' the
ring of polynomials
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
with coefficients in ''R''. Unless ''R'' is a field, ''R''
'X''is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.
* The
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
ring ''K''
''X''1,...,''X''''n''">''X''1,...,''X''''n'' over a field ''K'' (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if ''R'' is the localization of ''k''
'x'',''y'',''z''(''x''
2 + ''y''
3 + ''z''
7) at the
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
(''x'',''y'',''z'') then ''R'' is a local ring that is a UFD, but the formal power series ring ''R''
''X''">''X'' over ''R'' is not a UFD.
*The
Auslander–Buchsbaum theorem
In commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains.
The theorem was first proved by . They showed that regular local ring In abstract algebra, more specifically ring theory, l ...
states that every
regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...
is a UFD.
*