Does there exist a GCD domain $R$ in which the relationship $aR \cap b R \subseteq \mathrm{lcm}(a,b) R$ is untrue?
Showing that $(m)\cap (n)=(\operatorname{lcm}(m,n))$ and $(m)+(n)=(\gcd(m,n))$ for any $m,n\in\mathbb{Z}$