Suppose $I,J$ are ideals of a ring $R$. I know that $I+J$ is the smallest ideal containing $I\cup J$. Is $IJ$ the largest ideal contained in $I\cap J$?
For example, $2\Bbb Z$ and $3\Bbb Z$ are ideals of $\Bbb Z$. Their intersection $6\Bbb Z$ is itself an ideal of $\Bbb Z$. Their product $2\Bbb Z$$3\Bbb Z=6\Bbb Z$, which makes it the largest ideal contained in their intersection.
Is this true in general?
$I \cap J$ is the largest ideal contained in $I \cap J$.
In some situations $IJ = I \cap J$, but this definitely isn't always true. As a counterexample in the integers, take $I=J=2 \mathbb{Z}$. Then $I \cap J = 2 \mathbb{Z}$ but $IJ = 4 \mathbb{Z}$.
In the integers, the generator of $I \cap J$ is a least common multiple of the generators of $I$ and $J$, so you will have $I \cap J = IJ$ when $I$ and $J$ are relatively prime (i.e. $I+J=1$).