UFD implies GCDs exist

UFD implies GCDs exist: Let $R$ be a UFD and let $a,b\in R$ be not both zero. Then there exists $d\in R$ such that (i) $d\mid a$ and $d\mid b$ , and (ii) if $c\mid a$ and $c\mid b$ then $c\mid d$ . Any two such $d$ differ by multiplication by a unit; one writes $d=\gcd(a,b)$ .

Remarks

In a unique factorization domain, write $a$ and $b$ as products of prime elements and take the product of the common primes with minimal exponents to obtain a greatest common divisor. Uniqueness is up to associates, and in a UFD prime elements coincide with irreducible elements.