I'm working on the following problem in Ring Theory:
$\mathbb{Z}[x,y]/\left<y+1\right>$ is a UFD.
This made me wonder if it is possible to classify all ideals $I$ in $\mathbb{Z}[x,y]$ such that $\mathbb{Z}[x,y]/I$ is a UFD?
2026-03-29 22:28:31.1774823311
Find all ideals $\mathbb{Z}[x,y]$ such that the corresponding factor ring is a UFD
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This is an open problem. For example, one of the most famous problems in algebraic number theory is to identify the following set: $$\{ n \in \Bbb{N} : \Bbb{Z}[x] / \langle x^2- n \rangle \mbox{ is a UFD } \}$$ it is not even known whether this set is finite or infinite.
This means that your question is already very hard for the ring $\Bbb{Z}[x]$: much harder for $\Bbb{Z}[x,y]$.