prime ideals contains comaximal

Question

Let $R$ be a commutative ring with unity 1 and $I$, $J$ and $P$ ideals in $R$ show that if every prime ideal of $R$ contains either $I$ or $J$ ,but not both then $I$ and $J$ are comaximal site:math.ubc.ca

Answer 1

If $I$ and $J$ are not comaximal, then $I+J \neq R$ and so by Zorn lemma $$I+J \subset M$$ where $M$ is a maximal ideal. But a maximal ideal is also a prime ideal, and so we have a contradiction.