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Does a countable integral domain have only finitely many maximal ideals?

I've been thinking about this for awhile, I'd really appreciate a proof or counter example!! Thanks!

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    $\begingroup$ I think the integers are a counter-example. $\endgroup$ – B. Goddard Jun 7 at 22:19
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Not necessarily. A counterexample is $\mathbf Z$: its maximal ideals are generated by the primes and, as as you know, there's an infinity of them.

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