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Let $K$ be a number field and $\mathcal{O}_1, \mathcal{O}_2$ two orders in $K$. If $\mathcal{O}_1 \subset \mathcal{O}_2$, then is it always true that there is a containment of ring class fields for $\mathcal{O}_1$ and $\mathcal{O}_2$ in the other direction?

I thought about this question in two ways, but neither quite gets through the hoop. First, there's a lemma that says that if there is a containment between kernels of the Artin maps of two Abelian extensions with respect to a fixed modulus, then there is a containment of fields. However, in this case the two maps will not have the same modulus.

I suppose that an affirmative answer to my question implies that if $\mathcal{O}_1 \subset \mathcal{O}_2$, then $C(\mathcal{O}_2)$ is a quotient of $C(\mathcal{O}_1)$. This isn't quite clear to me: you're modding out by more principal ideals, but aren't you starting out with a different group of fractional ideals?

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