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?
Constructing a ring with a chain of ideals $(2) \subsetneq (2^{1/2}) \subsetneq (2^{1/3}) \subsetneq \cdots$
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abstract-algebra
ring-theory
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0It might be slightly trickier to prove that this ring does what you want. Do you see how to do it? – 2012-03-04
1 Answers
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Try $\mathbb{Z}[2^{\frac12},2^{\frac13},...]$.
This is a ring, and you have strict inclusions
$(0) \subsetneq (2) \subsetneq (\sqrt{2}) \subsetneq (2^{\frac 13}) \subsetneq ...$