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 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''">''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''">''X'' over ''R'' is not a UFD.
*The
Auslander–Buchsbaum theorem states that every
regular local ring is a UFD.
*