TITLE: Ideals in Dedekind domain
QUESTION [5 upvotes]: If I is a non-zero ideal in a Dedekind domain such that $I^m$ and $I^n$ are principal, and are equal to $(a)$ and $(b)$ respectively. How to show that $I^{(m,n)}$ is principal.
Try: $(m,n) = rm +sn$ So, $I^{(m,n)} = (a)^r(b)^s$,  where $r$, $s$ can be positive and negative. Both positive case is ok. but how to handle other cases.

REPLY [3 votes]: Denote by $e=(m,n)$. There exist $r, s>0$ such that $rm=sn+e$. 
Write $I^m=(a)$ and $I^n=(b)$, then $(a^r)=(b^s)I^e\subseteq (b^s)$. This implies in particular that $a^r=b^sc$ for some $c$ in your domain. Simplifying by $b^s$ (this is always possible in an integral domain), we get $(c)=I^e$. 
As said David in the comments, this holds in any integral domain.