I am looking at the definition (Sutherland, last page) of class group as the set of proper ideals of $\mathcal{O}$ modulo homothety, where:
- $\mathcal{O}$ is an order in an imaginary quadratic number field $K$
- An ideal $\mathfrak{a}$ is proper if and only if $\{\alpha \in K \mid \alpha \mathfrak{a} \subseteq \mathfrak{a} \} = \mathcal{O}$
- $\mathfrak{a}$ and $\mathfrak{b}$ are homothetic if and only if there exist $\alpha,\beta \in \mathcal{O}$ such that $\alpha\mathfrak{a} = \beta \mathfrak{b}$.
It's not clear to me why a product of proper ideals should be proper. According to Sutherland, it should be "clear". I also think that it's just a matter of some routine letter chasing, but I just can't prove it for some reason it. Is there something I am missing or is it not that trivial?