So far I have that if I + J = R, then 1 ∈ I and/or 1 ∈ J. Then I = R and/or J = R. If both I = J = R, then IJ = I ∩ J = R must be true because we already know that multiplying every element in I or J by every element in R will end up giving us R again. So IJ = I ∩ J = R.
If I = R, but J ≠ R, then IJ = J = I ∩ J because we already know that when all elements of J are multiplied by all elements of R, the result is J again. Therefore, IJ = I ∩ J = J.
Therefore, it is true that if I + J = R, then IJ = I ∩ J.
Does this proof work or am I not able to say that 1 ∈ I or 1 ∈ J?
Hint:
$IJ\subset I\cap J$ is trivial. For the reverse inclusion:
By hypothesis, there exist $u\in I$, $v\in J$ such that $u+v =1$. Now let $x\in I\cap J$. Use that $x=1\cdot x$.