Mathematical structure with greatest common divisors
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Rack and quandle
Quasigroup and loop
Abelian group
Magma
Lie group
Group theory
Ring-like
Ring
Rng
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Integral domain
Field
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Ring theory
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Total order
Heyting algebra
Boolean algebra
Map of lattices
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Vector space
Linear algebra
Algebra-like
Algebra
Associative
Non-associative
Composition algebra
Lie algebra
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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. Equivalently, any two elements of R have a least common multiple (LCM).[1]
A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian).
GCD domains appear in the following chain of class inclusions:
^Anderson, D. D. (2000). "GCD domains, Gauss' lemma, and contents of polynomials". In Chapman, Scott T.; Glaz, Sarah (eds.). Non-Noetherian Commutative Ring Theory. Mathematics and its Application. Vol. 520. Dordrecht: Kluwer Academic Publishers. pp. 1–31. doi:10.1007/978-1-4757-3180-4_1. MR 1858155.
In mathematics, a GCDdomain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique...
factorization domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains...
divisor of x and y is denoted gcd ( x , y ) {\displaystyle \gcd(x,y)} . For example, the GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4. In the name "greatest...
Dedekind domain, an integral domain in which every nonzero proper ideal factors into a product of prime ideals GCDdomain, an integral domain in which...
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...
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 (...
algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without...
integral domain is said to be pre-Schreier if every nonzero element is primal. A GCDdomain is an example of a Schreier domain. The term "Schreier domain" was...
{\displaystyle x\in \{0,1\}} . In a GCDdomain (for instance in Z {\displaystyle \mathbb {Z} } ), the operations of GCD and LCM are idempotent. In a Boolean...
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...
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...
binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor (GCD) of...
from the so-called "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization...
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...
element is an irreducible ideal. However, if D {\displaystyle D} is a GCDdomain and x {\displaystyle x} is an irreducible element of D {\displaystyle...
{\displaystyle \mathbb {Z} } is a Euclidean domain. This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain, and any positive integer can be...
a unique factorization domain). In particular, R {\displaystyle R} is also a Bézout domain, so it is a gcddomain and the gcd of any two elements satisfies...
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...
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...
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