Why $\langle I, J\rangle =R$ for distinct prime ideals $I$, $J$ of a principal ideal domain $R$?

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 1

Hint: Think of maximality (why does prime imply maximality in a PID?)

Answer 2

Without words:

$$1\in\langle I,J\rangle\implies\;\exists\,i\in I\,,\,j\in J\;\;s.t.\;\; i+j=1\implies$$

$$\forall\,x\in R\;,\;\;x=x\cdot1=x(i+j)=xi+xj\in\langle I,J\rangle$$