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

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.
The following diagram indicates how the different domains are related:
Euclidean domain  $\Longrightarrow$  PID  $\Longrightarrow$  UFD 

$\Downarrow$  $\Downarrow$  
Bezout domain  $\Longrightarrow$  gcd domain 
References
 1 D. D. Anderson, Advances in Commutative Ring Theory: Extensions of Unique Factorization, A Survey, 3rd Edition, CRC Press (1999)
 2 D. D. Anderson, NonNoetherian 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)
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1 not 0 by pahio ✘
please add to defines list by Mathprof ✓
gcd(0,0) = 0 by plclark ✓
GCD as ideal by pahio ✘