Unique Factorization Domain
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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 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 ...
''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 domains, hence all Euclidean domains, 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 ...
(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 integers 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 ring ''K'' ''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'' 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, ...
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. *\mathbb\left ^\right/math> is a UFD for all integers 1 ≤ ''n'' ≤ 22, but not for ''n'' = 23. *Mori showed that if the completion of a
Zariski ring In commutative algebra, a Zariski ring is a commutative Noetherian topological ring ''A'' whose topology is defined by an ideal \mathfrak a contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by under the ...
, such as a Noetherian local ring, is a UFD, then the ring is a UFD. The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of ''k'' 'x'',''y'',''z''(''x''2 + ''y''3 + ''z''5) at the prime ideal (''x'',''y'',''z''), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of ''k'' 'x'',''y'',''z''(''x''2 + ''y''3 + ''z''7) at the prime ideal (''x'',''y'',''z'') the local ring is a UFD but its completion is not. *Let R be a field of any characteristic other than 2. Klein and Nagata showed that the ring ''R'' 'X''1,...,''X''''n''''Q'' is a UFD whenever ''Q'' is a nonsingular quadratic form in the ''Xs and ''n'' is at least 5. When ''n''=4 the ring need not be a UFD. For example, R ,Y,Z,W(XY-ZW) is not a UFD, because the element XY equals the element ZW so that XY and ZW are two different factorizations of the same element into irreducibles. *The ring ''Q'' 'x'',''y''(''x''2 + 2''y''2 + 1) is a UFD, but the ring ''Q''(''i'') 'x'',''y''(''x''2 + 2''y''2 + 1) is not. On the other hand, The ring ''Q'' 'x'',''y''(''x''2 + ''y''2 – 1) is not a UFD, but the ring ''Q''(''i'') 'x'',''y''(''x''2 + ''y''2 – 1) is . Similarly the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
R 'X'',''Y'',''Z''(''X''2 + ''Y''2 + ''Z''2 − 1) of the 2-dimensional real sphere is a UFD, but the coordinate ring C 'X'',''Y'',''Z''(''X''2 + ''Y''2 + ''Z''2 − 1) of the complex sphere is not. *Suppose that the variables ''X''''i'' are given weights ''w''''i'', and ''F''(''X''1,...,''X''''n'') is a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
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 ''R'' 'X''1,...,''X''''n'',''Z''(''Z''''c'' − ''F''(''X''1,...,''X''''n'')) is a UFD .


Non-examples

*The
quadratic integer ring In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and (usual) integers. When algebrai ...
\mathbb Z
sqrt In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
/math> of all complex numbers 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 which 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 deno ...
of \mathbb Q
sqrt In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
/math> will fail to be a UFD unless d is a
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriza ...
. *The ring of formal power series over the complex numbers is a UFD, but the subring 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 fin ...
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 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 ...
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 z\in K ,y,z(z^2-xy) 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) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
and a
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bo ...
. Here, a greatest common divisor of ''a'' and ''b'' is an element ''d'' which
divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
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 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 ...
. * Any UFD is integrally closed. In other words, if R is a UFD with
quotient field In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
K, and if an element k in K is a
root In vascular plants, the roots are the 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 below the sur ...
of a
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\cd ...
with
coefficients In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
in R, then k is an element of R. * Let ''S'' be a
multiplicatively closed subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...
of a UFD ''A''. Then the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
S^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 lengt ...
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 example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
1
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 ...
is principal (a proof is given at the end). Also, a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
is a UFD if and only if its
ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
is trivial. In this case, it is in fact a principal ideal domain. 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 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 ...
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 pri ...
. ( Kaplansky) # ''A'' satisfies
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), and the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
''S''−1''A'' is a UFD, where ''S'' is a
multiplicatively closed subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...
of ''A'' generated by prime elements. (Nagata criterion) # ''A'' satisfies ACCP and every
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
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 In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
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 satisfying ACCP. # ''A'' 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 commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which a ...
in which every
divisorial ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
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.Bourbaki, 7.3, no 2, Theorem 1. 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) which is principal. By (2), the ring is a UFD.


See also

*
Parafactorial local ring In algebraic geometry, a Noetherian ring, Noetherian local ring ''R'' is called parafactorial if it has depth of a local ring, depth at least 2 and the Picard group Pic(Spec(''R'') − ''m'') of its spectrum (ring theory), spectrum with th ...
* Noncommutative unique factorization domain


Citations


References

* * Chap. 4. * Chapter II.5 of * * * {{Authority control Ring theory Algebraic number theory factorization