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If $A$ is a ring such that $A_{p}$ is a field for every prime ideal $p\subseteq A$, is $A$ a field?

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Yes, I would and perhaps even should and could. Would ya? Or in other words: what are your thoughts, your effort, your ideas on this? Note that the other direction of the claim is true, so this would be an iff claim... – DonAntonio Apr 3 '13 at 17:14

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The answer is no. Take $A$ to be a product of two fields. Notice that, here, $A$ isn't even a domain. So, we've shown not only that being a field isn't a local property, but being a domain isn't a local property, either.

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