Proof of Steinitz Theorem

Question

I want a source containing the proof of Steinitz Isomorphism Theorem stating:

For any Dedekind domain $R$ and any two nonzero ideals $I$ and $J$ of $R$ we have $I⊕J≅R⊕IJ$.

Thanks!

Answer 1

We have an exact sequence $0\to I\cap J\to I\oplus J\to I+J\to 0$. If $I+J=R$, then we are done.

Let's prove that there is an ideal $I'$ such that $I\simeq I'$ and $I'+J=R$. First chose an ideal $K$ such that $IK=aR$ and $K+J=R$. This can be done as follows: write $I=\prod p^{i_p}$, and for every $p$ appearing in $I$ and $J$ pick an element $x_p\in p^{i_p}-p^{i_p+1}$. From CRT there is $a\in R$ such that $a\equiv x_p\bmod p^{i_p+1}$ for all such $p$. We have $aR=IK$ with $K+IJ=R$.
Now write $K=\prod_{i=1}^tq_i^{v_i}$ and chose $b\in q_i^{v_i}-q_i^{v_i+1}$ for $i=1,\dots,t$ and $b\equiv 1\bmod J$. Then $bR=KI'$ with $I'+J=R$. Finally, $aI'=IKI'=bI$, so $I'\simeq I$.