Example of $IJ\not\subset(I+J)\cdot(I\cap J)$

Question

Can somebody give me an example of a case where $IJ\not\subset(I+J)\cdot(I\cap J)$ for $I,J$ ideals in a commutative ring $R$?

Answer 1

$\mathbb{Z}[x]$, e.g. with $I=(2)$ and $J=(x)$, then $$(I+J)(I\cap J)=(2,x)(2x)=(4x,2x^2)\subsetneq (2x)=IJ.$$