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Connected subset of Spec R

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Connected subset of Spec R

Does anyone know of an example of a commutative ring R s.t. Spec(R) has a subset X which is NOT connected, but the closure of X is?


This post is like 3 years ago. But I will answer this one, because it is easy :) Take any local ring R with Spec(R)=2normal-♯SpecR2\sharp Spec(R)=2 (for instance \Z2subscript\Z2\Z_{2}). Suppose MMM is the maximal ideal of RRR and then take the fiber product R×R/MRsubscriptRMRRR\times_{{R/M}}R then this ring has 3 ideals two of which are minimal and one is (the only) maximal ideal containing these two minimal prime ideal. If you take XXX to be the subset of Spec(R)SpecRSpec(R) containing these two minimal prime ideals then you have your example.

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