PlanetMath: gcd domain

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gcd domain

(Definition)

Let 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:

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

Any two gcd's of a pair of elements in are associates of each other. Therefore we can speak of the gcd of and with the knowledge that any two such gcd's are “equivalent” by a product of a unit. The formal way of doing this is to define $a\sim b$ iff and are associates, show that $\sim$ is an equivalence relation, and form the set of equivalence classes $D/\sim$ . Alternatively, we can define

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

and denote $\gcd(a,b)$ to be any element of $\operatorname{GCD}(a,b)$ .

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

An integral domain is called a gcd domain if any two elements of , not both zero, have a gcd.

Remarks

A unique factorization domain, or UFD is a gcd domain, but the converse is not true.
A Bezout domain is always a gcd domain. A gcd domain is a Bezout domain if $\gcd(a,b) = ra+sb$ for any $a, b \in D$ and some $r, s \in D$ .
In a gcd domain, an irreducible element is a prime element.
A gcd domain is integrally closed. In fact, it is a Schreier domain.