Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.


"Easy" means that I can explain it to my undergrad students, although I will be happy with any example.



An example of two elements without a greatest common divisor
[Here's](http://en.wikipedia.org/wiki/Greatest_common_divisor) an example stolen blatantly from wikipedia.


Let $R=\mathbb{Z}[\sqrt{-3}]$, let $a=4=2\*2=(1+\sqrt{-3})(1-\sqrt{-3})$ and $b=2(1+\sqrt{-3})$. Now, $2$ and $1+\sqrt{-3}$ are both maximal among divisors, but are not associates, thus, there is not GCD.


