Examples of PID's with finitely many maximal ideals

Question

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.

Answer 1

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 $S^{-1}M_1, S^{-1}M_2, \dots , S^{-1}M_k$.