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 ...
, 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
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, ...
holds. Specifically, a UFD 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 ...
(a
nontrivial commutative ring
In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
in which the product of any two non-zero elements is non-zero) in which every non-zero non-
unit element 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, 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 formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
s in one or more variables with coefficients coming from the integers or from a
field.
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, 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 ...
''R'' in which every non-zero element ''x'' of ''R'' which is not a unit can be written 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 ''p''
''i'' of ''R'':
: ''x'' = ''p''
1 ''p''
2 ⋅⋅⋅ ''p''
''n'' with
and this representation is unique in the following sense:
If ''q''
1, ..., ''q''
''m'' are irreducible elements of ''R'' such that
: ''x'' = ''q''
1 ''q''
2 ⋅⋅⋅ ''q''
''m'' with ,
then , and there exists a
bijective map such that ''p''
''i'' is
associated to ''q''
''φ''(''i'') for .
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 (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
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 Euclidean division of integers. Th ...
s, are UFDs. In particular, the
integers
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 ...
(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 is prime or can be represented uniquely as a product of prime numbers, ...
''), 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 ...
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 = \frac ...
s are UFDs.
* If ''R'' is a UFD, then so is ''R''
'X'' the
ring of polynomials 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 su ...
ring 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
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
Arts, entertainment, and media
* ''Local'' (comics), a limited series comic book by Bria ...
. For example, if ''R'' is the localization of at the
prime ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
then ''R'' is a local ring that is a UFD, but the formal power series ring ''R'' over ''R'' is not a UFD.
* The
Auslander–Buchsbaum theorem states that every
regular local ring is a UFD.
*
coordinate ring
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.
More formally, an affine algebraic set is the set of the common zeros over an algeb ...
of the 2-dimensional
real sphere is a UFD, but the coordinate ring of the complex sphere is not.
* Suppose that the variables ''X''
''i'' are given weights ''w''
''i'', and is a
homogeneous polynomial of weight ''w''. Then if ''c'' is coprime to ''w'' and ''R'' is a UFD and either every
finitely generated projective module over ''R'' is
free or ''c'' is 1 mod ''w'', the ring is a UFD.
Non-examples
* The
quadratic integer ring
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree tw ...
\mathbb Z sqrt/math> of all complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s of the form
a+b\sqrt, where ''a'' and ''b'' are integers, is not a UFD because 6 factors as both 2×3 and as
\left(1+\sqrt\right)\left(1-\sqrt\right). These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3,
1+\sqrt, and
1-\sqrt are
associate. It is not hard to show that all four factors are irreducible as well, though this may not be obvious. See also ''
Algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
''.
* For a
square-free positive integer ''d'', the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
of
\mathbb Q sqrt/math> will fail to be a UFD unless ''d'' is a Heegner number
In number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from int ...
.
* The ring of formal power series over the complex numbers is a UFD, but the
subring
In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
of those that converge everywhere, in other words the ring of
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
s in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.:
*:
\sin \pi z = \pi z \prod_^ \left(1-\right).
Properties
Some concepts defined for integers can be generalized to UFDs:
* In UFDs, every
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 ...
is
prime
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 ...
. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the
ACCP is a UFD if and only if every irreducible element is prime.
* Any two elements of a UFD have a
greatest common divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...
and a
least common multiple
In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...
. Here, a greatest common divisor of ''a'' and ''b'' is an element ''d'' that
divides both ''a'' and ''b'', and such that every other common divisor of ''a'' and ''b'' divides ''d''. All greatest common divisors of ''a'' and ''b'' are
associated.
* Any UFD is
integrally closed. In other words, if ''R'' is a UFD with
quotient field ''K'', and if an element ''k'' in ''K'' is a
root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
of a
monic polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
with
coefficients
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a ...
in ''R'', then ''k'' is an element of ''R''.
* Let ''S'' be a
multiplicatively closed subset of a UFD ''A''. Then the
localization ''S''
−1''A'' is a UFD. A partial converse to this also holds; see below.
Equivalent conditions for a ring to be a UFD
A
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
integral domain is a UFD if and only if every
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For an example of vertical extent, "This basketball player is 7 foot 1 inches in height." For an e ...
1
prime ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
is principal (a proof is given at the end). Also, a
Dedekind domain is a UFD if and only if its
ideal class group
In mathematics, the ideal class group (or class group) of an algebraic number field K is the quotient group J_K/P_K where J_K is the group of fractional ideals of the ring of integers of K, and P_K is its subgroup of principal ideals. The ...
is trivial. In this case, it is in fact a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
.
In general, for an integral domain ''A'', the following conditions are equivalent:
# ''A'' is a UFD.
# Every nonzero
prime ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
of ''A'' contains 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 ...
.
# ''A'' satisfies
ascending chain condition on principal ideals (ACCP), and the
localization ''S''
−1''A'' is a UFD, where ''S'' is a
multiplicatively closed subset of ''A'' generated by prime elements. (Nagata criterion)
# ''A'' satisfies
ACCP and every
irreducible is
prime
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 ...
.
# ''A'' is
atomic and every
irreducible is
prime
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 ...
.
# ''A'' is a
GCD domain
In mathematics, a GCD domain (sometimes called just domain) is an integral domain ''R'' with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated ...
satisfying
ACCP.
# ''A'' is a
Schreier domain,
[A Schreier domain is an integrally closed integral domain where, whenever ''x'' divides ''yz'', ''x'' can be written as so that ''x''1 divides ''y'' and ''x''2 divides ''z''. In particular, a GCD domain is a Schreier domain] and
atomic.
# ''A'' is a
pre-Schreier domain and
atomic.
# ''A'' has a
divisor theory in which every divisor is principal.
# ''A'' is a
Krull domain in which every
divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.)
# ''A'' is a Krull domain and every prime ideal of height 1 is principal.
In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since every prime ideal is generated by a prime element in a PID.
For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains a height one prime ideal (induction on height) that is principal. By (2), the ring is a UFD.
See also
*
Parafactorial local ring
*
Noncommutative unique factorization domain
Citations
References
*
*
*
* Chap. 4.
* Chapter II.5
*
*
*
*
{{Authority control
Ring theory
Algebraic number theory
factorization