I don't understand a point in the proof of the theorem 8.4 in Anthony Scholl's Number FIelds lecture notes.
Theorem: Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and and $I \subset \mathcal{O}_K$ a non-zero ideal. Then there exists a non-zero ideal $J$ such that $IJ = (\alpha)$ for some $\alpha \in \mathcal{O}_K$.
Proof: By induction on $N(I)$. Let $\alpha \in I$ be non-zero. By earlier lemma $\exists \beta \in \mathcal{O}_K \setminus (\alpha)$ such that $\beta I \subset (\alpha)$. So $\alpha^{-1} \beta I \subset \mathcal{O}_K$. On the other hand we also have $\alpha ^{-1} \beta \notin \mathcal{O}_K$ and hence $\alpha ^{-1} \beta I \not\subset I$ by another earlier lemma and so $I' := I + (\alpha^{-1} \beta)$ is an ideal that strictly contains $I$. By induction hypothesis say $I'J'=(\gamma)$ for some $\gamma$ and let $J=(\alpha , \beta ) J'$. Then $IJ=I (\alpha , \beta ) J'=\alpha I' J' = (\alpha \gamma)$.
What I don't understand is the last line of the proof - where does $I (\alpha , \beta ) J'=\alpha I' J'$ come from?
To me it seems that $(\alpha, \beta) I = \alpha I + \beta I$ whereas $\alpha I' = \alpha I + (\beta)$ so the second ideal is strictly greater.
I suspect $I'$ should be defined as $I+\alpha^{-1}\beta I$. Then $I'$ is an ideal of $\mathcal{O}_K$ as both $I$ and $\alpha^{-1}\beta I$ are. Also $\alpha I'=\alpha I+\beta I$, which is what you want.