When an Intersection of Prime Ideals is a Prime Ideal
Solution 1:
If $P_i \subset P_j$ for every $j$, then $P_1 \cap \cdots \cap P_n = P_i$ is prime. Conversely, if $P_1 \cap \cdots \cap P_n = Q$ is prime then $Q \subset P_j$ for every $j$ and also there exists some $i$ such that $P_i \subset Q$. Hence $Q=P_i$. Here i used the fact that if $a$ is an ideal such that $a \supset p_1 p_2 \cdots p_m$ where the $p_i$ are prime, then $a$ contains some $p_k$. (If i am not mistaken, this is proposition 1.11b in Atiyah-MacDonald.)