What about R=K[[x]] (formal power series), K a field? It is a local ring and not a field. Hence it is not von Neumann regular and therefore its spectrum is not extremelly disconnected. On the other hand R is a domain and hence its only minimal prime is (0). The minSpec is a singleton and ...
Have I missed anything?
Hi Carriegos,
We can see that SpecR is extremally disconnected iff R is a Baer-ring(i.e., The annihilator of any ideal is generated by an idempotent). Also, for a reduced ring R, R is a Baer-ring iff R[X] is a Baer ring. So K[x] is Baer-ring and hence SpecK[X] is an extremally disconnected space.
Hello,
My proposal is formal power series, not polynomials.
On the other hand, a question: Set K=C={Complex numbers} in your reply. Then following your reply Spec(C[x]) is extremelly disconnected.
But Spec(C[x]) is the affine one-dimensional space. Surely I've missed something ... somewhere.
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