gcd domain

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

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

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

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

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

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

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

If $\mathrm{GCD}(a,b)$ 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 $a,b\in D$, $\mathrm{GCD}(a,b)\ne \mathrm{\varnothing }$.

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 $D$ 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.

• Given an integral domain, one can similarly define an lcm of two elements $a,b$: it is an element $c$ such that $a\mid c$ and $b\mid c$, and if $d$ is an element such that $a\mid d$ and $b\mid d$, then $c\mid d$. Then, alcm domain is an integral domain such that every pair of elements has a lcm. As it turns out, the two notions are equivalent: an integral domain is lcm iff it is gcd.

The following diagram indicates how the different domains are related:

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