[Math] Products of ideals is an ideal and comaximal ideals

Question

1. Let $I$ and $J$ be two ideals in a ring $R$. Prove that $IJ$ is an ideal.

2. Prove that if $R$ is a commutative ring with two ideals satisfying $I+J=R$ then $IJ=I\cap J$.

I could prove that $IJ$ has an identity, inverse as viewed over addition. I have also prived left and right multiplication property but I am unable to proof the closure over addition.

Answer 1

Hint (for 2). $1=a+b$ $\Rightarrow$ $x=ax+bx$.