What are examples of non-PID rings for which all nonzero prime ideal are maximal? I am not so sure about the non-examples of wiki. Is $\mathbb{Z}[X]$ such an example?
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Bill Dubuque
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roi_saumon
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Is it possible to classify such commutative rings ? – reuns Oct 31 '18 at 21:06
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2any Dedekind domain which it is not PID. – Mustafa Oct 31 '18 at 21:14
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See here for all prime ideals maximal, i.e. Krull dim $0$ (vs. $,\le 1$ here). $\ \ $ – Bill Dubuque Mar 22 '25 at 19:17
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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.
Cehiju
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