TITLE: When the localization of a ring is a field
QUESTION [15 upvotes]: Let $R$ be a commutative noetherian ring with no nonzero nilpotents. Let $p$ be a minimal prime of $R$. Could you help me to prove that $R_p$ is a field?

REPLY [4 votes]: Maybe a little bit more clear than the first answer:
$R_p$ itself is reduced and local. Moreover $\dim R_p=0$, so we can ask the following: is it true that a local reduced ring with dimension 0 is a field? Answer: yes! Reduced means that the nilradical of $R$ is $(0)$, but in this case the nilradical is the only prime ideal of $R$, so the only prime ideal of $R$ is (0) which means that $R$ is a field.