EDIT: Since the original question was to vague I will pose some stronger conditions:
Given $n \in \mathbb{N}$ with $n \geq 3$ I want to know if there is a ring $R$ with the following properties:
- $\mathbb{Z}[\sqrt{-n}] \subset R$
- $R$ is a UFD but not a field (in particular I want that there are only finitely many integers $k \in \mathbb{Z}$ that are invertible in $R$)
- For any $\alpha \in R$ there is a unit $\eta \in R^*$ such that $\alpha \eta \in \mathbb{Z}[\sqrt{-n}]$
If $n=3$, one can take $R$ to be the ring of integers of $\mathbb{Q}(\sqrt{-n})$. This does not work for $n \geq 4$, since then the ring of integers of $\mathbb{Q}(\sqrt{-n})$ has only two units. Are there other rings $R$ that could work?
Original question: Let $n \in \mathbb{N}, n \geq 3$. Then $\mathbb{Z}[\sqrt{-n}]$ is not a UFD. I want to know which extensions $\mathbb{Z}[\sqrt{-n}]$ are (or can be) a UFD. Two examples would be:
- For some $n$, the ring of integers of $\mathbb{Q}[\sqrt{-n}]$ is a UFD and this is an extension of $\mathbb{Z}[\sqrt{-n}]$.
- Every field extension of $\mathbb{Z}[\sqrt{-n}]$ (for example $\mathbb{Q}(\sqrt{-n})$) is a UFD
Are there any other extensions of $\mathbb{Z}[\sqrt{-n}]$ (for example using localization, adjoining other algebraic numbers, other concepts, or combining some of these) that are UFD?
