Question: Let $R$ be a principal ideal domain with identity and $I$, $J$ be distinct prime ideals of $R$. Prove that $1 \in \langle I, J\rangle$ hence $\langle I, J\rangle = R$. How to prove?

Answer: 
Hint: Think of maximality (why does prime imply maximality in a PID?)
Let $I=\langle p\rangle$ and $J=\langle q\rangle$ be two distinct prime ideals in the PID $R$. Therefore neither $p\mid q$ nor $q\mid p$.

Consider the ideal $\langle I,J\rangle=\langle c\rangle$. Since $p,q\in\langle c\rangle$ we have $c\mid p,q$, and since primes are irreducible, then either $c$ is $p$ times a unit (which is impossible since it would imply $p\mid q$), or $q$ times a unit (which is also impossible) or a unit itself, and therefore $\langle c\rangle=R$.