Math 2.0

Commutative algebra Definition

Greatest common divisor in an integral domain

Builds on Divisibility in a commutative ringAssociate elementsIntegral domain

Definition

Let RR be an integral domain and let a,b∈Ra, b \in R. An element d∈Rd \in R is a greatest common divisor of aa and bb if it has these two properties.

  1. d∣ad \mid a and d∣bd \mid b.
  2. For every c∈Rc \in R with c∣ac \mid a and c∣bc \mid b, c∣dc \mid d.

Remarks

If dd and d′d' are two greatest common divisors of aa and bb, then dd and d′d' are associates. In ℤ\mathbb{Z}, the greatest common divisors of 1212 and 1818 are 66 and −6-6. A greatest common divisor need not exist. A GCD domain is an integral domain in which every pair of elements has one. In a principal ideal domain, dd is a greatest common divisor of aa and bb if and only if (a,b)=(d)(a, b) = (d).

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDivisibility in a commutative ring, definition, no evidence levelDivisibility in a com…definitionIntegral domain, definition, no evidence levelIntegral domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionGreatest common divisor in an integral domain, definition, no evidence levelGreatest common divis…definition
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Unit of a ring definition
  5. Zero divisor definition
  6. Divisibility in a commutative ring definition
  7. Integral domain definition
  8. Associate elements definition
  9. Greatest common divisor in an integral domain definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/GCD_domain, lead section
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MSC13A05
DefinesGreatest common divisor in an integral domain
UsesDivisibility in a commutative ring, Associate elements, Integral domain

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