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Type of integral domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which
Unique_factorization_domain
(Mathematical) decomposition into a product
example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful
Factorization
In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property. The ring of Hurwitz quaternions
Noncommutative unique factorization domain
Noncommutative_unique_factorization_domain
About products of primitive polynomials
integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem
Gauss's_lemma_(polynomials)
Integers have unique prime factorizations
unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be represented uniquely
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Algebra with unique prime factorization
such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that
Dedekind_domain
Commutative ring with no zero divisors other than zero
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Integral_domain
Commutative ring with a Euclidean division
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Euclidean_domain
Algebraic structure
Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all
Principal_ideal_domain
Mathematical structure with greatest common divisors
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
GCD_domain
Polynomial without nontrivial factorization
in unique factorization domains. The polynomial ring F[x] over a field F (or any unique-factorization domain) is again a unique factorization domain. Inductively
Irreducible_polynomial
integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive
Primitive_part_and_content
Integral domain in which the sum of two principal ideals is again a principal ideal
generated ideals; if so, it is not a unique factorization domain (UFD), but is still a GCD domain. The theory of Bézout domains retains many of the properties
Bézout_domain
Algebraic structure with addition and multiplication
is a unique factorization domain, then R[t] is a unique factorization domain. Finally, R is a field if and only if R[t] is a principal ideal domain. Let
Ring_(mathematics)
Algebraic structure
integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed
Integrally_closed_domain
Complex number whose real and imaginary parts are both integers
they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties
Gaussian_integer
In number theory, measure of non-unique factorization
a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they
Ideal_class_group
Atomic domains are different from unique factorization domains in that this decomposition of an element into irreducibles need not be unique; stated
Atomic_domain
In algebra, element without non-trivial factors
non-unit element are uniquely defined, up to the multiplication by a unit, then the integral domain is called a unique factorization domain, but this does not
Irreducible_element
Algebraic structure
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Commutative_ring
Commutative ring with a well behaved theory of prime factorization
{\displaystyle B} is a Krull domain. Any unique factorization domain is a Krull domain. Conversely, a Krull domain is a unique factorization domain if (and only if)
Krull_ring
Algebraic structure
integral domains. If R is a unique factorization domain then the same holds for R[X]. This results from Gauss's lemma and the unique factorization property
Polynomial_ring
Greatest common divisor of polynomials
The proof that a polynomial ring over a unique factorization domain is also a unique factorization domain is similar, but it does not provide an algorithm
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Number divisible only by 1 and itself
hold for unique factorization domains. The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example
Prime_number
Concept in abstract algebra
to multiplication by units). R {\displaystyle R} is a unique factorization domain with a unique irreducible element (up to multiplication by units). R
Discrete_valuation_ring
Mathematical ring with well-behaved ideals
domain R, every element can be factorized into irreducible elements (in short, R is a factorization domain). Thus, if, in addition, the factorization
Noetherian_ring
Branch of algebra
their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring. Algebraic geometry
Ring_theory
Finite extension of the rationals
field is not necessarily a principal ideal domain, and not necessarily even a unique factorization domain. The Gaussian rationals, denoted Q ( i ) {\displaystyle
Algebraic_number_field
Topics referred to by the same term
divisor Principal ideal domain, an integral domain in which every ideal is principal Unique factorization domain, an integral domain in which every non-zero
Domain
Mathematical concept
In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that
Square-free_element
Mathematical structure where elements are primal
in a pre-Schreier domain, every irreducible is prime. In particular, an atomic pre-Schreier domain is a unique factorization domain; this generalizes
Schreier_domain
Rational numbers with root 5 added
the ring of its algebraic integers is a principal ideal domain and a unique factorization domain. Any positive element of the golden field can be written
Golden_field
Analogue of a prime number in a commutative ring
elements from irreducible elements, a concept that is the same in unique factorization domains but not the same in general. An element p of a commutative ring
Prime_element
Sufficient condition for polynomial irreducibility
decompositions of axn are possible in (Z/pZ)[x], which is a unique factorization domain. In particular the constant terms of G and H vanish in the reduction
Eisenstein's_criterion
Algorithm for computing greatest common divisors
domain: Any number from a Euclidean domain can be factored uniquely into irreducible elements. Any Euclidean domain is a unique factorization domain (UFD)
Euclidean_algorithm
Positive real number which when multiplied by itself gives 5
example of an integral domain that is not a unique factorization domain. For example, the number 6 has two inequivalent factorizations within this ring: 6
Square_root_of_5
{\displaystyle \mathbb {Q} ({\sqrt {-7}})} . This ring is a unique factorization domain. Eisenstein integer Gaussian integer Conway, John Horton; Smith
Kleinian_integer
Submodule of fractions in abstract algebra
integers O K {\displaystyle {\mathcal {O}}_{K}} is from being a unique factorization domain (UFD). This is because h K = 1 {\displaystyle h_{K}=1} if and
Fractional_ideal
Computational method
the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm
Factorization_of_polynomials
Negative integer two units from the origin in mathematics
[{\sqrt {d}}]} is a unique factorization domain, all numbers in Q [ d ] {\displaystyle \mathbb {Q} [{\sqrt {d}}]} have a unique factorization. For example,
−2
Algebraic structure where all polynomials have roots
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Algebraically_closed_field
Branch of number theory
that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors.
Algebraic_number_theory
Smallest positive number divisible by two integers
elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple
Least_common_multiple
Algebraic construction
{-5}})(1-{\sqrt {-5}}).} A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals. The units of a ring of integers
Ring_of_integers
Type of mathematical expression
polynomials. If the coefficients belong to a field or a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication
Polynomial
Type of ring in commutative algebra
Auslander–Buchsbaum theorem states that every regular local ring is a unique factorization domain. Every localization, as well as the completion, of a regular
Regular_local_ring
Generalizations of codimension-1 subvarieties of algebraic varieties
positive integer. Since the polynomial ring k[x1, ..., xn] is a unique factorization domain, the divisor class group of affine space An over k is equal to
Divisor_(algebraic_geometry)
Algebraic structure
polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field may be factored in a unique way (up to the order of the
Finite_field
Number in {..., –2, –1, 0, 1, 2, ...}
\mathbb {N} } is called a bijection. Mathematics portal Canonical factorization of a positive integer Complex integer Hyperinteger Integer complexity
Integer
Abstract algebra concept
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
Field_of_fractions
Largest integer that divides given integers
integral domains. However, if R is a unique factorization domain or any other GCD domain, then any two elements have a GCD. If R is a Euclidean domain in which
Greatest_common_divisor
Function defined on integers in number theory
computed. The arithmetic derivative can also be extended to any unique factorization domain (UFD), such as the Gaussian integers and the Eisenstein integers
Arithmetic_derivative
Algebraic ring without a multiplicative identity
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Rng_(algebra)
Structure-preserving function between two rings
ring (the ring whose only element is zero). For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial
Ring_homomorphism
generate the unit ideal. unique factorization domain Also called a factorial domain. A unique factorization domain is an integral domain such that every element
Glossary of commutative algebra
Glossary_of_commutative_algebra
Algebraic theorem
Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains. The theorem was first proved by Maurice Auslander and David
Auslander–Buchsbaum_theorem
Subset of a ring that forms a ring itself
Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3. Sharpe, David (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.
Subring
Theorem on polynomial roots modulo prime powers
factors modulo p into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of p (the case of roots corresponds
Hensel's_lemma
Equivalence class of objects sharing local properties at a point in a topological space
&x\neq 0,\\0,&x=0.\end{cases}}} This ring is also not a unique factorization domain. This is because all UFDs satisfy the ascending chain condition
Germ_(mathematics)
Mathematical structure in abstract algebra
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
*-algebra
Function in algebra
ideal domain, K be its field of fractions, and π be an irreducible element of R. Since every principal ideal domain is a unique factorization domain, every
Valuation_(algebra)
Algebra based on a vector space with a quadratic form
}}v\in V} (where 1A denotes the multiplicative identity of A), there is a unique algebra homomorphism f : B → A such that the following diagram commutes
Clifford_algebra
principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably unique factorization domains and left or
Ascending chain condition on principal ideals
Ascending_chain_condition_on_principal_ideals
German polymath and scholar (1777–1855)
[ i ] {\displaystyle \mathbb {Z} [i]} , showed that it is a unique factorization domain, and generalized some key arithmetic concepts, such as Fermat's
Carl_Friedrich_Gauss
Condition under which an odd prime is a sum of two squares
the fact that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since p ∈ Z does not divide either of the
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Algorithms for matrix decomposition
non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more
Non-negative matrix factorization
Non-negative_matrix_factorization
Method in computational algebra
(recalling that the ring of polynomials over a finite field is a unique factorization domain). All possible factors of f ( x ) {\displaystyle f(x)} are contained
Berlekamp's_algorithm
Scientific area at the interface between computer science and mathematics
reducing polynomials (such as those over a ring of integers or a unique factorization domain) to a variant efficiently computable via a Euclidean algorithm
Computer_algebra
On prime factors of integer products
that triangles are congruent. In general, to show that a domain is a unique factorization domain, it suffices to prove Euclid's lemma and the ascending
Euclid's_lemma
Root of a quadratic polynomial with a unit leading coefficient
for every Dedekind domain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain. This occurs if and
Quadratic_integer
Algebraic structure used in analysis
real or complex numbers, there is a corresponding connected Lie group, unique up to covering spaces (Lie's third theorem). This correspondence allows
Lie_algebra
Conditions under which the congruence x^3 equals p (mod q) is solvable
^{2}\right\}.} Z [ ω ] {\displaystyle \mathbb {Z} [\omega ]} is a unique factorization domain. The primes fall into three classes: 3 is a special case: 3 =
Cubic_reciprocity
Theory of a class of elliptic curves
\mathbf {Z} \left[{\frac {1+{\sqrt {-163}}}{2}}\right]} is a unique factorization domain. Here ( 1 + − 163 ) / 2 {\displaystyle (1+{\sqrt {-163}})/2}
Complex_multiplication
Free object in the category of associative algebras
X_{j_{m}},} and the product of two arbitrary R-module elements is thus uniquely determined (because the multiplication in an R-algebra must be R-bilinear)
Free_algebra
In mathematics, dimension of a ring
ring is an example of such a ring. A Noetherian integral domain is a unique factorization domain if and only if every height 1 prime ideal is principal
Krull_dimension
Ratio of polynomial functions
. However, since F [ X ] {\displaystyle F[X]} is a unique factorization domain, there is a unique representation for any rational expression P / Q {\displaystyle
Rational_function
Branch of mathematics
formulated the Gaussian integers and showed that they form a unique factorization domain (UFD) and proved the biquadratic reciprocity law. Jacobi and
Abstract_algebra
Field in algebraic number theory
{\displaystyle E} . If the ring of integers of K {\displaystyle K} is a unique factorization domain, in particular if K = Q {\displaystyle K=\mathbb {Q} } , then
Hilbert_class_field
Generalization of vector spaces from fields to rings
not be the same for all bases (that is to say that they may not have a unique rank) if the underlying ring does not satisfy the invariant basis number
Module_(mathematics)
Unique ring consisting of one element
integer n, the ring Z/nZ is a domain if and only if n is prime, but 1 is not prime. For each ring A, there is a unique ring homomorphism from A to the
Zero_ring
Fully simplified fraction
unique prime factorization of integers, since a/b = c/d implies ad = bc, and so both sides of the latter must share the same prime factorization,
Irreducible_fraction
Matrix normal form
are unique since any PID is also a unique factorization domain). In particular, R {\displaystyle R} is also a Bézout domain, so it is a gcd domain and
Smith_normal_form
In algebra, expression of an ideal as the intersection of ideals of a specific type
p_{r}^{d_{r}}\rangle .} Similarly, in a unique factorization domain, if an element has a prime factorization f = u p 1 d 1 ⋯ p r d r , {\displaystyle
Primary_decomposition
and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q
List of number fields with class number one
List_of_number_fields_with_class_number_one
Algebraic ring that need not have additive negative elements
Y,Z]/(XZ-Y^{2})} demonstrates independence of some statements about factorization true in N {\displaystyle \mathbb {N} } . There are P A {\displaystyle
Semiring
Every nonzero proper ideal in the ring of integers of a number field factorizes uniquely
admits unique factorization into a product of nonzero prime ideals. In other words, every ring of integers of a number field is a Dedekind domain. Keith
Fundamental theorem of ideal theory in number fields
Fundamental_theorem_of_ideal_theory_in_number_fields
Topics referred to by the same term
electoral coalition (French: Union des forces démocratiques) Unique factorization domain, in abstract algebra United Front Department, a North Korean
UFD
Submodule of a mathematical ring
generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory)
Ideal_(ring_theory)
Branch of algebra that studies commutative rings
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Commutative_algebra
Ring that is also a vector space or a module
product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map m : A ⊗ R A → A {\displaystyle m:A\otimes _{R}A\to A} . The
Associative_algebra
Counterintuitive mathematical object
Some number fields have rings of integers that do not form a unique factorization domain, for example the extended field Q ( − 5 ) {\displaystyle \mathbb
Pathological_(mathematics)
Elements taken to zero by a homomorphism
domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to
Kernel_(algebra)
Infinite sum that is considered independently from any notion of convergence
series exists. It turns out that if A has a multiplicative inverse, it is unique, and we denote it by A−1. Now we can define division of formal power series
Formal_power_series
Fraction with denominator a power of two
subtraction of more general fractions involves integer multiplication and factorization to reach a common denominator. Therefore, dyadic fractions can be easier
Dyadic_rational
Algebraic structure
converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie's theorem
Noncommutative_ring
Ring ideal generated by a single element of the ring
ideal domain (PID) is an integral domain in which every ideal is principal. Any PID is a unique factorization domain; the normal proof of unique factorization
Principal_ideal
Branch of mathematics that studies algebraic structures
Integral domain, Domain (ring theory) Field of fractions, Integral closure Euclidean domain, Principal ideal domain, Unique factorization domain, Dedekind
List of abstract algebra topics
List_of_abstract_algebra_topics
equivalent to hyperplane divisors. Since the ring of polynomials is a unique factorization domain, any prime ideal of height 1 is principal, which shows that any
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Topological structure in number theory
complete Noetherian regular local ring, and in particular a unique factorization domain. It follows from the Weierstrass preparation theorem for formal
Iwasawa_algebra
Set without nontrivial polynomial equalities
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Algebraic_independence
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