TITLE: Non-integral scheme having integral local rings
QUESTION [33 upvotes]: I can show that if $X$ is a scheme such that all local rings $\mathcal{O}_{X,x}$ are integral and such that the underlying topological space is connected and Noetherian, then $X$ is itself integral. 
This doesn't seem to work without the "Noetherian" condition. But can anyone think about a nice counterexample to illustrate this? So I am looking for a non-integral scheme - with connected underlying topological space - having integral local rings.

REPLY [9 votes]: Hochster has an elegant construction which associates a commutative ring to each infinite  totally ordered set with the property that strictly between two distinct elements there is a third one. 
The spectrum of such a ring is a connected affine scheme of dimension one, all the local rings of which are domains. The ring itself, however is NOT a domain.
So, every ordered set with the property mentioned above yields a scheme with the required property.
Here is the link to Hochster's (one-page) construction
http://www.math.lsa.umich.edu/~hochster/614F08/ECdom.sol.pdf