Let $R$ be a Dedekind domain. If $I$,$J$ are $R$-ideals, prove that there exists an $\alpha\in I$ such that $\gcd(\langle\alpha\rangle,IJ)=I$.
Well if I could prove that there existed an $\alpha\in I$ and $j\in J$ such that $\alpha r_1 +jr_2=1$, then I would have $I\supseteq\gcd(\langle\alpha\rangle,IJ)=\langle\alpha\rangle+IJ\supseteq I\langle\alpha\rangle+IJ = I(\langle\alpha\rangle+J)=IR=I$. But who knows if that is the case. Honestly I have no idea what to do here, I tried looking at some examples but ideals in $\mathbb{Z}$ are too trivial and finding $\alpha$ in $\mathbb{Z}[\sqrt{-5}]$ is too complex (no pun intended). Hopefully someone can help me with this, thanks.