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There are absolutely flat rings $A$ (i.e. every $A$-module is flat) with nonisolated (automatically closed) points $x$ in the spectrum. Then take the inclusion $x\to \mathrm{Spec}(A)$. Examples of such rings are ($k$ being any field):

• infinite products of fields, e.g. $k^\mathbb{N}$: you get a nontrivial maximal ideal from any nonprincipal ultrafilter on $\mathbb{N}$,

• the ring of locally constant $k$-valued functions on any infinite compact (infinite) totally disconnected compact Hausdorff space $Y$: then it is a classical exercise that the spectrum is $Y$.

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There are absolutely flat rings $A$ (i.e. every $A$-module is flat) with nonisolated (automatically closed) points $x$ in the spectrum. Then take the inclusion $x\to \mathrm{Spec}(A)$. Examples of such rings are ($k$ being any field):

• infinite products of fields, e.g. $k^\mathbb{N}$: you get a nontrivial maximal ideal from any nonprincipal ultrafilter on $\mathbb{N}$,

• the ring of locally constant $k$-valued functions on any infinite compact totally disconnected space $Y$: then it is a classical exercise that the spectrum is $Y$.