A commutative ring $R$ is said to be a PF-ring if every principal ideal of $R$ is a flat $R$-module. Also, a prime ideal $P$ of $R$ to be a Krull associated prime of $R$ if for every element $x\in P$ , there exists $y\in R$ such that $x \in ann_R(y)\subseteq P$, where $ann_R(y):=\{r\in R\mid ry=0_R\}$.

I am looking for a PF-ring with a Krull associated prime ideal $P$ such that $P$ is not a minimal prime ideal of $R$, that is, there exists a prime ideal $Q$ of $R$ with $Q\subseteq P$ and $Q\not= P$.


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