\begin{definition}[Definition:Greatest Common Divisor/Integral Domain]
Let $\struct {D, +, \times}$ be an integral domain whose zero is $0$.
Let $a, b \in D: a \ne 0 \lor b \ne 0$.
Let $d \divides a$ denote that $d$ is a divisor of $a$.
Let $d \in D$ have the following properties:
: $(1): \quad d \divides a \land d \divides b$
: $(2): \quad c \divides a \land c \divides b \implies c \divides d$
Then $d$ is called '''a greatest common divisor of $a$ and $b$''' (abbreviated '''GCD''' or '''gcd''') and denoted $\gcd \set {a, b}$.
That is, in the integral domain $D$, $d$ is the GCD of $a$ and $b$ {{iff}}:
: $d$ is a common divisor of $a$ and $b$
: Any other common divisor of $a$ and $b$ also divides $d$.
We see that, trivially:
:$\gcd \set {a, b} = \gcd \set {b, a}$
so the set notation is justified.
\end{definition}