$ \mathbf{Z}[x]$ is not PID.
we know $\mathbb Z$ is a Unique Factorization Domain, so $\mathbb Z[x]$ is UFD, but why isn't it PID (since I think $\mathbb Z$ is PID)?
2026-04-26 01:10:09.1777165809
Why $ \mathbb{Z}[x]$ is not Principal Ideal Domain
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If it were a PID, then every nonzero prime ideal would be maximal. But $\mathbb Z[X]/p \mathbb Z[X] = \mathbb (Z/p\mathbb Z)[X]$ is an integral domain which is not a field.