Least common multiples exist in a PID

Let $R$ be a principal ideal domain. Prove that any two nonzero elements in $R$ have a least common multiple.

We saw in this previous exercise that the least common multiples of $a$ and $b$ are precisely the generators of the $\subseteq$-largest principal ideals contained in $(a) \cap (b)$. Since $R$ is a principal ideal domain, $(a) \cap (b) = (\ell)$ is itself principal, so that a least common multiple indeed exists.

• DL  On November 28, 2011 at 9:28 pm

There is some typo,,, 🙂

• nbloomf  On November 29, 2011 at 12:05 am

Thanks!