Uniqueness of greatest common divisor

Question

Suppose $R$ is an integral domain and $a,b\in R$ have no common factors other than units. Then $1$ is a greatest common divisor of $a$ and $b$.

Is $1$ the only greatest common divisor, or can there be others?

Answer 1

The GCD is very rarely unique. If $x$ is a GCD of $a$ and $b$, then so is any associate of $x$, that is, any number of the form $ux$ with $u$ a unit. This is easily verifiable from the definition

$gcd(a,b)=x$ if $x$ is a common divisor of $a$ and $b$ and is divisible by all other common divisors of $a$ and $b$

Thus the GCD is only unique when you have a ring with one unit. In your particular case, every unit is a GCD.