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Do you know some examples of principal ideal domains which have finitely many maximal ideals? More generally, do you know how to build such domains? I don't look for fields and discrete valuation rings.

Thank you.

Ángel Valencia
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1 Answers1

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An easy way to construct a PID with finitely many maximal ideals.

  1. Start with a PID $R$.

  2. Pick finitely many $M_1, M_2, \dots , M_k$ maximal ideals of $R$, and call $S=R \setminus (M_1 \cup M_2 \cup \dots \cup M_k)$: this is a multiplicative subset of $R$.

  3. $S^{-1}R$ is a PID (because it is a localization of a PID) whose maximal ideals are exactly $S^{-1}M_1, S^{-1}M_2, \dots , S^{-1}M_k$.

Crostul
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