What are examples of non-PID rings for which all prime ideal are maximal? I am not so sure about the non-examples of wiki. Is $\mathbb{Z}[X]$ such an example?
2026-03-27 13:39:54.1774618794
non-PID which where all prime ideals are maximal
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$\mathbb{Z}[x]$ is not a PID, but the ideal $(2)$ is prime and not maximal. The ring $k[x]/(x^2)$ is an example of a ring with exactly one prime ideal, but is not a domain, so not a PID. In $k[x,y]/(x,y)^2$ you again have exactly one prime ideal, and it is not principal.