In a Dedekind domain every ideal is either principal or generated by two elements.

Question

Prove that in a Dedekind domain every ideal is either principal or generated by two elements.

Help me some hints.

Thanks a lot!

Answer 1

Hints:

Let $R$ be a Dedekind domain, and let $I\subseteq R$ be any ideal. Factor $I$ as $\mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_n^{e_n}$. Take ANY $\alpha\in I-\{0\}$ and factor $(\alpha)$ as $\mathfrak{p}_1^{f_1}\cdots\mathfrak{p}_n^{f_n}\mathfrak{q}_1^{g_1}\cdots\mathfrak{q}_m^{g_m}$. We know that $f_i\geqslant e_i$ since $(\alpha)\subseteq I$ so that $I\mid (\alpha)$. Now explain why there exists some $\beta\in R$ such that $v_{\mathfrak{p}_i}(\beta)=e_i$ and $v_{\mathfrak{q}_i}(\beta)=0$. Explain then why

$$(\alpha,\beta)=\text{gcd}((\alpha),(\beta))=\mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_n^{e_n}=I$$

This, in fact, shows that for any $I$ and any $\alpha\in I-\{0\}$ you can always find a complementary generator.