Greatest common divisor in an integral domain

Definition

Let $R$ be an integral domain and let $a, b \in R$. An element $d \in R$ is a greatest common divisor of $a$ and $b$ if it has these two properties.

1. $d \mid a$ and $d \mid b$.
2. For every $c \in R$ with $c \mid a$ and $c \mid b$, $c \mid d$.

Remarks

If $d$ and $d'$ are two greatest common divisors of $a$ and $b$, then $d$ and $d'$ are associates. In $\mathbb{Z}$, the greatest common divisors of $12$ and $18$ are $6$ and $-6$. A greatest common divisor need not exist. A GCD domain is an integral domain in which every pair of elements has one. In a principal ideal domain, $d$ is a greatest common divisor of $a$ and $b$ if and only if $(a, b) = (d)$.

Explanations

L1 human

In a general ring, "greatest" does not refer to size. It means the largest for divisibility: every other common divisor divides it. For the integers, this gives the usual gcd, up to the sign.