I was looking for a specific ring with infinitely many prime ideals such that they are totally ordered by inclusion. A valuation ring with rank $\mathbb{N} \cup \infty$ should work, or something like that, but I don't know any.

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    $\begingroup$ @JeskoHüttenhain Probably the meaning is that the set of prime ideals is totally ordered by inclusion. $\endgroup$ – egreg Nov 27 '18 at 23:33
  • $\begingroup$ @egreg that makes perfect sense, thanks for helping me out there. $\endgroup$ – Jesko Hüttenhain Nov 27 '18 at 23:34
  • $\begingroup$ Do you mean that there is some set of prime ideals which is totally ordered by inclusion or that the collection of all prime ideals needs to be totally ordered by inclusion? Also, $\mathbb{N} \cup \{ \infty \}$ isn't a group under addition. $\endgroup$ – Qiaochu Yuan Nov 28 '18 at 8:45
  • $\begingroup$ Thanks, edited, I mean all primes. $\endgroup$ – Kato Nov 28 '18 at 11:08
  • $\begingroup$ @JyrkiLahtonen But there are other prime ideals. $\endgroup$ – egreg Nov 28 '18 at 11:26

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