# Example of a principal prime ideal containing a proper prime non-zero ideal. [duplicate]

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 $$$$0 \subsetneq \mathfrak{p} \subsetneq \left\langle p \right\rangle$$$$