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

Now, a gcd of two elements is in general not unique. However, 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 $$\operatorname{GCD}(a,b):=\lbrace c\in D\mid c\mbox{ is a gcd of }a\mbox{ and }b\rbrace,$$ and, if there is no confusion, 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.

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, a lcm 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.