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It is true that if $A$ is UFD and $I=\left\langle p \right\rangle$ is a principal prime ideal, it does not exist a proper prime non-zero ideal contained in $I$, but this property does not hold for general rings (commutative rings with 1). I would appreciate some examples, that is, a ring $A$ and prime ideals $\mathfrak{p}, \left\langle p \right\rangle$ such that \begin{equation} 0 \subsetneq \mathfrak{p} \subsetneq \left\langle p \right\rangle \end{equation}

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marked as duplicate by user26857 commutative-algebra Oct 16 '18 at 17:42

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