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I am trying to construct a ring that contains this chain of principal ideals: $(2)\subsetneq (2^{1/2})\subsetneq (2^{1/3})\subsetneq \cdots$ How can I show that it gives a ring?

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    It might be slightly trickier to prove that this ring does what you want. Do you see how to do it?2012-03-04

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Try $\mathbb{Z}[2^{\frac12},2^{\frac13},...]$.

This is a ring, and you have strict inclusions

$(0) \subsetneq (2) \subsetneq (\sqrt{2}) \subsetneq (2^{\frac 13}) \subsetneq ...$