$\newcommand{\ann}{\operatorname{ann}}$I am looking for a reduced commutative ring $R $ with $1$ contacting an ideal $I $ such that there exists a minimal prime ideal $p $ of $R $ with $\ann (I) \subseteq p $ and $\ann (\ann (I)) \subseteq p$.
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1$\begingroup$ Quick observation: this would imply that $ann(P)\subseteq ann(I)\cap ann(ann(I))$, and by reducedness $ann(P)=\{0\}$. If I remember correctly, some well-known results say that minimal primes of Noetherian rings are never faithful. So far, I do not have any candidates for a ring like this, if it exists. It would probably be good, while constructing, to make $P$ infinitely generated. $\endgroup$– rschwiebMar 20, 2017 at 15:06
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