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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 in which a statement analogous to the fundamental theorem of arithmetic 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 s ...
(a nontrivial
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 ...
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'' (a ...
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 elements), 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. 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 s ...
''R'' in which every non-zero element ''x'' of ''R'' can be written as a product (an empty product if ''x'' is a unit) of irreducible elements ''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'' (a ...
''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 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 principa ...
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 (also see fundamental theorem of arithmetic), the Gaussian integers 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 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 s ...
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 wi ...
(''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 states that every regular local ring 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, 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 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 ideal ...
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 In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characteriz ...
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 \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. * 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 factori ...
. *The ring of formal power series over the complex numbers is a UFD, but the
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
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 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 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 ...
. 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 b ...
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 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 su ...
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^+\ ...
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 of a UFD ''A''. Then the localization 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 ab ...
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 wi ...
is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group 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 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 principa ...
. 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 wi ...
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 (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 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 by two given elements. Equivalen ...
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 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.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 * Noncommutative unique factorization domain


Citations


References

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