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There is a nice classification of prime ideals in the ring $\mathbb{Z}[x]$, see this question.

Is there any generalization of this result, on $\mathbb{Z}[x_1,\cdots,x_n]$?

Due to this post, I guess there is not a 'theorem' for large $n$. However, does anyone know the answer for $n=2$?

What are the prime ideals of $\mathbb{Z}[x,y]$?

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