Let $D$ be an u.f.d and $p(x),q(x) \in D[x]$ is it necessary that:
lcm$(p(x),q(x))\cdot\gcd(p(x),q(x)) = up(x)q(x)$? Where $u \in D$ is invertible.
I know this fact is true if $D[x]=F[x]$ where $F$ is a field and the proof of that required p.i.d so my bet is that this doesnt hold for any d.f.u however I can't come up with a counter-example.