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Mathematical structure with greatest common divisors
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 minimum
GCD_domain
Type of integral domain
factorization domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains
Unique_factorization_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
Algebraic structure
integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed
Principal_ideal_domain
About products of primitive polynomials
any GCD domain (an integral domain over which greatest common divisors exist). In particular, a polynomial ring over a GCD domain is also a GCD domain. If
Gauss's_lemma_(polynomials)
Largest integer that divides given integers
The GCD is a commutative function: gcd(a, b) = gcd(b, a). The GCD is an associative function: gcd(a, gcd(b, c)) = gcd(gcd(a, b), c). Thus gcd(a, b,
Greatest_common_divisor
Algebraic structure
integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed
Integrally_closed_domain
Topics referred to by the same term
Dedekind domain, an integral domain in which every nonzero proper ideal factors into a product of prime ideals GCD domain, an integral domain in which
Domain
Integral domain in which the sum of two principal ideals is again a principal ideal
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 of PIDs,
Bézout_domain
Algebraic structure with addition and multiplication
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Ring_(mathematics)
Commutative ring with a Euclidean division
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Euclidean_domain
Greatest common divisor of polynomials
factorization domain. If c is any common divisor of p and q, then c divides their GCD. gcd ( p , q ) = gcd ( q , p ) . {\displaystyle \gcd(p,q)=\gcd(q,p).} gcd (
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Algorithm for computing greatest common divisors
algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder
Euclidean_algorithm
Algebraic structure
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Commutative_ring
In algebra, element without non-trivial factors
element is an irreducible ideal. However, if D {\displaystyle D} is a GCD domain and x {\displaystyle x} is an irreducible element of D {\displaystyle
Irreducible_element
Number in {..., –2, –1, 0, 1, 2, ...}
\mathbb {Z} } is a Euclidean domain. This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain, and any positive integer can be
Integer
Property of operations
{\displaystyle x\in \{0,1\}} . In a GCD domain (for instance in Z {\displaystyle \mathbb {Z} } ), the operations of GCD and LCM are idempotent. In a Boolean
Idempotence
Mathematical structure where elements are primal
integral domain is said to be pre-Schreier if every nonzero element is primal. A GCD domain is an example of a Schreier domain. The term "Schreier domain" was
Schreier_domain
Algebraic structure
geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials
Polynomial_ring
Algebraic construction
Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain. The ring
Ring_of_integers
Generalization of vector spaces from fields to rings
realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces;
Module_(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
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
Submodule of a mathematical ring
Bézout's identity, we have n Z + m Z = gcd ( n , m ) Z . {\displaystyle n\mathbb {Z} +m\mathbb {Z} =\operatorname {gcd} (n,m)\mathbb {Z} .} Let R = C [ x
Ideal_(ring_theory)
Free object in the category of associative algebras
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Free_algebra
Branch of functional analysis
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Operator_algebra
Smallest positive number divisible by two integers
| b | gcd ( a , b ) = | b | | a | gcd ( a , b ) , {\displaystyle \operatorname {lcm} (a,b)=|a|\,{\frac {|b|}{\gcd(a,b)}}=|b|\,{\frac {|a|}{\gcd(a,b)}}
Least_common_multiple
Method for computing the relation of two integers with their greatest common divisor
common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that a x + b y = gcd ( a , b ) {\displaystyle
Extended_Euclidean_algorithm
Algebraic ring without a multiplicative identity
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Rng_(algebra)
Algorithm for computing the greatest common divisor
{\displaystyle \gcd(2u,2v)=2\cdot \gcd(u,v)} : 2 {\displaystyle 2} is a common divisor. gcd ( u , 2 v ) = gcd ( u , v ) {\displaystyle \gcd(u,2v)=\gcd(u,v)} if
Binary_GCD_algorithm
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
Algebraic ring that need not have additive negative elements
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Semiring
Branch of algebra
as follows: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring Algebraic geometry is in many
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
Algebraic structure used in analysis
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Lie_algebra
Structure-preserving function between two rings
is a maximal ideal of R. If R and S are commutative and S is an integral domain, then ker(f) is a prime ideal of R. If R and S are commutative, S is a field
Ring_homomorphism
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
Reduction of a ring by one of its ideals
distinct from the "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization
Quotient_ring
Matrix normal form
a unique factorization domain). In particular, R {\displaystyle R} is also a Bézout domain, so it is a gcd domain and the gcd of any two elements a ,
Smith_normal_form
Ring built from other rings (mathematics)
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Product_of_rings
Branch of number theory
adopts the definition of unique factorization used in unique factorization domains (UFDs). In a UFD, the prime elements occurring in a factorization are only
Algebraic_number_theory
Subset of a ring that forms a ring itself
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Subring
Fraction with denominator a power of two
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Dyadic_rational
Algebra over a field where binary multiplication is not necessarily associative
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Non-associative_algebra
Algebraic structure
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Semifield
(Mathematical) decomposition into a product
integral domain in which greatest common divisors exist (known as a GCD domain) is a UFD. Every principal ideal domain is a UFD. A Euclidean domain is an
Factorization
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
Ring that is also a vector space or a module
analog of Levi's theorem for Lie algebras. Let R be a Noetherian integral domain with field of fractions K (for example, they can be Z, Q). A lattice L in
Associative_algebra
Tensor product of algebras over a field; itself another algebra
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Tensor_product_of_algebras
Mathematical term in group theory
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Prüfer_group
Submodule of fractions in abstract algebra
of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals
Fractional_ideal
Algebra based on a vector space with a quadratic form
on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford correlation. The
Clifford_algebra
Infinite sum that is considered independently from any notion of convergence
version of the Hilbert basis theorem), if R {\displaystyle R} is an integral domain, then so is R [ [ X ] ] {\displaystyle R[[X]]} , and if K {\displaystyle
Formal_power_series
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
Unique ring consisting of one element
two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor
Zero_ring
Study of numbers that are not solutions of polynomials with rational coefficients
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Transcendental_number_theory
an integral domain. Then the following are equivalent. A is a UFD. A satisfies (ACCP) and every irreducible of A is prime. A is a GCD domain satisfying
Ascending chain condition on principal ideals
Ascending_chain_condition_on_principal_ideals
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Free product of associative algebras
Free_product_of_associative_algebras
Analogue of a prime number in a commutative ring
integral domain, every prime is irreducible but the converse is not true in general. However, in unique factorization domains, or more generally in GCD domains
Prime_element
Relating two numbers and their greatest common divisor
always produces one of these two minimal pairs. Let a = 12 and b = 42, then gcd (12, 42) = 6. Then the following Bézout's identities are displayed with the
Bézout's_identity
Concept in mathematical ring theory
generic magma with divisibility between every pair of elements Zero divisor GCD domain In this article, rings are assumed to have a 1. Bourbaki 1989, p. 97 Bourbaki
Divisibility_(ring_theory)
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)
Construction within abstract algebra
construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds
Total_ring_of_fractions
Use of functions that call themselves
: gcd ( x , y ) = gcd ( y , x % y ) {\displaystyle \gcd(x,y)=\gcd(y,x\%y)} if y ≠ 0 {\displaystyle y\neq 0} gcd ( x , 0 ) = x {\displaystyle \gcd(x,0)=x}
Recursion_(computer_science)
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Semiprimitive_ring
Special case of colimit in category theory
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Direct_limit
Category whose objects are rings and whose morphisms are ring homomorphisms
include the full subcategories of commutative rings, integral domains, principal ideal domains, and fields. The category of commutative rings, denoted CRing
Category_of_rings
Mathematical concept
In mathematics, an overring of an integral domain contains the integral domain, and the integral domain's field of fractions contains the overring. Overrings
Overring
Branch of mathematics
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Press, ISBN 0-226-42454-5, MR 0345945 Picavet, Gabriel (1999), "About GCD domains", in Dobbs, David E. (ed.), Advances in commutative ring theory. Proceedings
Goldman_domain
ring of Gaussian integers m+ni. GCD 1. Abbreviation for greatest common divisor 2. A GCD domain is an integral domain such that any two elements have
Glossary of commutative algebra
Glossary_of_commutative_algebra
Integers have unique prime factorizations
The canonical representations of the product, greatest common divisor (GCD), and least common multiple (LCM) of two numbers a and b can be expressed
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Polynomial with no repeated root
the GCD computation of the input polynomial and its derivative. More precisely, if T n {\displaystyle T_{n}} is the time needed to compute the GCD of two
Square-free_polynomial
contents: c ( gcd ( P 1 , P 2 ) ) = gcd ( c ( P 1 ) , c ( P 2 ) ) . {\displaystyle c(\operatorname {gcd} (P_{1},P_{2}))=\operatorname {gcd} (c(P_{1})
Primitive_part_and_content
Algebraic structure
composition ring does not have a multiplicative unit. If R is an integral domain, the field R(X) of rational functions also has a substitution operation
Composition_ring
Division with remainder of integers
{\displaystyle R,} with m > 0 {\displaystyle m>0} and gcd ( R , m ) = 1 , {\displaystyle \gcd(R,m)=1,} let R − 1 {\displaystyle R^{-1}} be the modular
Euclidean_division
Special type of element of a set
Domain Operation Absorber real numbers ⋅ {\displaystyle \cdot } multiplication 0 integers gcd {\displaystyle \gcd } greatest common divisor 1 n {\displaystyle
Absorbing_element
About simultaneous modular congruences
{n_{k}}},\end{aligned}}} has a solution if and only if gcd ( n i , n j ) {\displaystyle \gcd(n_{i},n_{j})} divides a i − a j {\displaystyle a_{i}-a_{j}}
Chinese_remainder_theorem
a b ] = [ gcd ( a , b ) 0 ] . {\displaystyle {\begin{bmatrix}s&t\\u&v\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}={\begin{bmatrix}\gcd(a,b)\\0\end{bmatrix}}
Linear_equation_over_a_ring
Semiring defined over probabilities
for long sequences, so it is common to perform computations in the log domain. Implementing the Viterbi algorithm in log-space means using log {\displaystyle
Viterbi_semiring
Algorithm for public-key cryptography
algorithm, since lcm(a, b) = |ab|/gcd(a, b). λ(n) is kept secret. Choose an integer e such that 1 < e < λ(n) and gcd(e, λ(n)) = 1; that is, e and λ(n)
RSA_cryptosystem
Method in computational algebra
successively compute GCDs of the form above until we find a non-trivial factor. Since the ring of polynomials over a field is a Euclidean domain, we may compute
Berlekamp's_algorithm
Algorithm for factoring polynomials over finite fields
fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by David G. Cantor and Hans Zassenhaus in 1981
Cantor–Zassenhaus_algorithm
Declarative, general-purpose programming language
following Gödel module is a specification of the greatest common divisor (GCD) of two numbers. It is intended to demonstrate the declarative nature of
Gödel_(programming_language)
Function whose domain is the positive integers
{ 1 if gcd ( a , n ) = 1 , 0 if gcd ( a , n ) ≠ 1. {\displaystyle \chi _{0}(a)={\begin{cases}1&{\text{if }}\gcd(a,n)=1,\\0&{\text{if }}\gcd(a,n)\neq
Arithmetic_function
Periodic set of points
which is equivalent to the coordinates being coprime, gcd ( a 1 , … , a n ) = 1 {\displaystyle \gcd(a_{1},\ldots ,a_{n})=1} . Every one-dimensional sublattice
Lattice_(group)
Digital signal generator
(GRR) given by GRR = 2 N GCD ( Δ F , 2 N ) {\displaystyle {\mbox{GRR}}={\frac {2^{N}}{{\mbox{GCD}}(\Delta F,2^{N})}}} where GCD is the greatest common divisor
Numerically controlled oscillator
Numerically_controlled_oscillator
Genus of flowering plants in the daisy family
Flann, Christina (ed.). "Search Calendula". Global Compositae Database (GCD). Archived from the original on 24 September 2020. Retrieved 31 March 2011
Calendula
1928 novel by Erich Maria Remarque
Complete List". The New York Times. ISSN 0362-4331. Retrieved March 13, 2023. "GCD :: Issue :: Classics Illustrated #95 [O] – All Quiet on the Western Front"
All Quiet on the Western Front
All_Quiet_on_the_Western_Front
Comic book series
"GCD :: Series :: Solo Ex-Mutants". "GCD :: Series :: The New Humans". "GCD :: Series :: Wild Knights". "GCD :: Series :: Shattered Earth". "GCD ::
Ex-Mutants
may denote the greatest common divisor of a and b. Notation gcd ( a , b ) {\displaystyle \gcd(a,b)} is often used instead. (□, □, □) If x, y, z are vectors
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Condition under which an odd prime is a sum of two squares
when q > 2 {\displaystyle q>2} is even, gcd ( a , q / 2 ) = 1 {\displaystyle (a,q/2)=1} ; otherwise since gcd ( a , q / 2 ) ∣ q / 2 ∣ q ∣ a 2 + b 2 {\displaystyle
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Matrix form in linear algebra
Havas, George; Majewski, Bohdan S.; Matthews, Keith R. (1998). "Extended GCD and Hermite normal form algorithms via lattice basis reduction". Experimental
Hermite_normal_form
2002 video game
2024. "Freedom Force (Volume)". Comic Vine. Retrieved January 28, 2022. "GCD :: Issue :: Freedom Force #6". www.comics.org. Retrieved January 28, 2022
Freedom Force (2002 video game)
Freedom_Force_(2002_video_game)
Fictional character created by Robert E. Howard
Rodriguez "Conan (comic book character)". Comicvine.com. Retrieved 2012-11-17. "GCD :: Series :: Cuentos de Abuelito". comics.org. Swaine, Jon (7 November 2008)
Conan_the_Barbarian
1862 narrative poem by Christina Rossetti
archived from the original on 12 December 2021, retrieved 18 January 2019 "GCD :: Issue :: Dare #1". www.comics.org. Retrieved 25 January 2019. Davenport
Goblin_Market
Type of character
March 26, 2016.{{cite web}}: CS1 maint: deprecated archival service (link) "GCD :: Issue :: Thrilling Comics #v1#2 (2)". Comics.org. January 11, 1940. Archived
Superhero
Special-purpose integer factorization algorithm
factorizations of n {\displaystyle n} : n = gcd ( a + b , n ) ⋅ gcd ( a − b , n ) {\displaystyle n=\gcd(a+b,n)\cdot \gcd(a-b,n)} . If done right, it is almost
Special_number_field_sieve
Complex number whose real and imaginary parts are both integers
and b are both integers. As for any unique factorization domain, a greatest common divisor (gcd) of two Gaussian integers a, b is a Gaussian integer d that
Gaussian_integer
Ring ideal generated by a single element of the ring
{\mathrm {gcd} } (a,b)\rangle ,} by induction on the number of generators it follows that I {\displaystyle I} is principal. Any Euclidean domain is a PID;
Principal_ideal
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GCD DOMAIN
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