I was working in David Cox's Primes of the Form $x^2 + ny^2$, and am currently stuck on this question:
Let $K$ be a quadratic number field and $\mathcal{O}_K$ its ring of integers. An $\mathcal{O}_K$-ideal $\mathfrak{a}$ is said to be proper if the set $\{\beta \in K : \beta \mathfrak{a} \subset \mathfrak{a}\} = \mathcal{O}_K$. Show that every ideal of $\mathcal{O}_K$ is proper.
I figure I will have to make use of the fact that $\mathcal{O}_K$ is the maximal order, or possibly the fact that $\mathcal{O}_K$ is a Dedekind domain, but am not quite sure how the argument will go. Any hints would be greatly appreciated!
Edit: I should mention that the containment $\supset$ is immediate because $\mathfrak{a}$ is an ideal.
If $\mathfrak{a}$ is an ideal of $\mathcal{O}_K$, then $\{\beta \in K : \beta \mathfrak{a} \subseteq \mathfrak{a}\}$ is an order that contains $\mathcal{O}_K$. Now if $\mathcal{O}_K$ is the maximal order...