GCD domain

Remarks

Here $x \mid y$ means divisibility in $R$; a greatest common divisor is unique up to multiplication by a unit. Example: $\Z$ is a GCD domain, and so is every unique factorisation domain, where a gcd is read off from the factorizations. Non-example: in $\Z[\sqrt{-5}]$, with the norm $N(x + y\sqrt{-5}) = x^2 + 5y^2$, the elements $6$ and $2(1 + \sqrt{-5})$ have no gcd. Indeed $2$ and $1 + \sqrt{-5}$ are common divisors, so the norm of a gcd $d$ would be divisible by $N(2) = 4$ and by $N(1 + \sqrt{-5}) = 6$. Also $N(d)$ would divide $N(6) = 36$ and $N(2 + 2\sqrt{-5}) = 24$, so $N(d) = 12$. But $x^2 + 5y^2 = 12$ has no integer solution, so clause 2 fails for every common divisor $d$. Equivalent form: any two elements have a least common multiple.