On Prime and Maximal Ideals in a Commutative Ring with Unity

Question

Let $R$ be a commutative ring with $1 \neq 0$, $I$ and $P$ are ideals of $R$. If $P$ is prime and $I \cap P \neq 0$, does it follows that either $I \subseteq P$ or $I$ is also a prime ideal incomparable to $P$? Does it also extend over the case where $P$ is maximal?

Answer 1

The ring $\mathbb{Z}$ already has counterexamples - for instance consider $P=(2)$ and $I=(15)$, we have $I\cap P=(30)$ and $P$ is maximal, but $I\not\subseteq P$ and $I$ is not a prime ideal.