gcd domain

Throughout this entry, let D be a commutative ring with 10.

A gcd (greatest common divisorMathworldPlanetmathPlanetmath) of two elements a,bD, is an element dD such that:

  1. 1.

    da and db,

  2. 2.

    if cD with ca and cb, then cd.

For example, 0 is a gcd of 0 and 0 in any D. In fact, if d is a gcd of 0 and 0, then d0. But 00, so that 0d, which means that, for some xD, d=0x=0. As a result, 0 is the unique gcd of 0 and 0.

In general, however, a gcd of two elements is not unique. For example, in the ring of integersMathworldPlanetmath, 1 and -1 are both gcd’s of two relatively prime elementsMathworldPlanetmath. By definition, any two gcd’s of a pair of elements in D are associatesMathworldPlanetmath of each other. Since the binary relationMathworldPlanetmath “being associates” of one anther is an equivalence relationMathworldPlanetmath (not a congruence relationPlanetmathPlanetmath!), we may define the gcd of a and b as the set

GCD(a,b):={cDc is a gcd of a and b},

For example, as we have seen, GCD(0,0)={0}. Also, for any aD, GCD(a,1)=U(D), the group of units of D.

If there is no confusion, let us denote gcd(a,b) to be any element of GCD(a,b).

If GCD(a,b) contains a unit, then a and b are said to be relatively prime. If a is irreduciblePlanetmathPlanetmath, then for any bD, a,b are either relatively prime, or ab.

An integral domainMathworldPlanetmath D is called a gcd domain if any two elements of D, not both zero, have a gcd. In other words, D is a gcd domain if for any a,bD, GCD(a,b).


  • A unique factorization domainMathworldPlanetmath, or UFD is a gcd domain, but the converseMathworldPlanetmath is not true.

  • A Bezout domain is always a gcd domain. A gcd domain D is a Bezout domain if gcd(a,b)=ra+sb for any a,bD and some r,sD.

  • In a gcd domain, an irreducible element is a prime element.

  • A gcd domain is integrally closedMathworldPlanetmath. In fact, it is a Schreier domain.

  • Given an integral domain, one can similarly define an lcm of two elements a,b: it is an element c such that ac and bc, and if d is an element such that ad and bd, then cd. Then, a lcm domain is an integral domain such that every pair of elements has a lcm. As it turns out, the two notions are equivalentMathworldPlanetmathPlanetmathPlanetmath: an integral domain is lcm iff it is gcd.

The following diagram indicates how the different domains are related:

Euclidean domainMathworldPlanetmath (http://planetmath.org/EuclideanRing) PID UFD
Bezout domain gcd domain


  • 1 D. D. Anderson, Advances in Commutative Ring Theory: ExtensionsPlanetmathPlanetmath of Unique Factorization, A Survey, 3rd Edition, CRC Press (1999)
  • 2 D. D. Anderson, Non-Noetherian Commutative Ring Theory: GCD Domains, Gauss’ Lemma, and Contents of Polynomials, Springer (2009)
  • 3 D. D. Anderson (editor), Factorizations in Integral Domains, CRC Press (1997)
Title gcd domain
Canonical name GcdDomain
Date of creation 2013-03-22 14:19:51
Last modified on 2013-03-22 14:19:51
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 26
Author CWoo (3771)
Entry type Definition
Classification msc 13G05
Related topic GreatestCommonDivisor
Related topic BezoutDomain
Related topic DivisibilityInRings
Defines gcd
Defines greatest common divisor
Defines relatively prime
Defines lcm domain