TITLE: Locally a domain and connected implies a domain
QUESTION [5 upvotes]: Let $R$ be a commutative ring with unit. Let $R_p$ be a domain for all $p\in SpecR$ and let $SpecR$ be connected. Is it true that $R$ is a domain or can someone provide a counterexample. Note here that $R$ is not necessarily a Noetherian ring. For a Noetherian ring this is easy.

REPLY [3 votes]: See https://mathoverflow.net/questions/7477/non-integral-scheme-having-integral-local-rings