Ideals in Dedekind domain 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.

Even in the other cases the argument works, you just have fractional ideals instead. Viewing $I$ as an element $\overline{I}$ of the ideal class group of $A$ your question can be stated as: Suppose $\overline{I}^m=\overline{I}^n=0$ in the ideal class group, then $\overline{I}^{(m,n)}=0$. This statement is true in any group.
