# gcd domain

Throughout this entry, let $D$ be a commutative ring with $1\neq 0$.

A gcd (greatest common divisor   ) of two elements $a,b\in D$, is an element $d\in D$ such that:

1. 1.

$d\mid a$ and $d\mid b$,

2. 2.

if $c\in D$ with $c\mid a$ and $c\mid b$, then $c\mid d$.

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 $d\mid 0$. But $0\mid 0$, so that $0\mid d$, which means that, for some $x\in D$, $d=0x=0$. 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 integers  , $1$ and $-1$ are both gcd’s of two relatively prime elements  . By definition, any two gcd’s of a pair of elements in $D$ are associates  of each other. Since the binary relation  “being associates” of one anther is an equivalence relation  (not a congruence relation  !), we may define the gcd of $a$ and $b$ as the set

 $\operatorname{GCD}(a,b):=\{c\in D\mid c\mbox{ is a gcd of }a\mbox{ and }b\},$

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

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

If $\operatorname{GCD}(a,b)$ contains a unit, then $a$ and $b$ are said to be relatively prime. If $a$ is irreducible  , then for any $b\in D$, $a,b$ are either relatively prime, or $a\mid b$.

An integral domain  $D$ 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 $a,b\in D$, $\operatorname{GCD}(a,b)\neq\varnothing$.

Remarks

The following diagram indicates how the different domains are related:

 Euclidean domain  (http://planetmath.org/EuclideanRing) $\Longrightarrow$ PID $\Longrightarrow$ UFD $\Downarrow$ $\Downarrow$ Bezout domain $\Longrightarrow$ gcd domain

## References

• 1 D. D. Anderson, , 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