I am looking for an example of ring $A$, where the Euclid lemma is true. But, the ring A contains two elements $a$ and $b$ with non gcd, and lcm.
I know that a notherian ring is factorial iff the Euclid's lemma is valid. Thus, $A$ mustn't be noetherian.
The ring of the entire complex functions, and the ring of algebraic integers, aren't noetherian, satisfy Euclid's lemma, but all pair of elements have a gcd and lcm....
Edit : The version of the euclid lemma that I have in mind is : if $p$ is irreductible and $p$ divides the product ab then $p$ divides $a$ or $p$ divide $b$.