Product of two ideals doesn't equal the intersection The product of two ideals is defined as the set of all finite sums $\sum  f_i  g_i$, with $f_i$ an element of $I$, and $g_i$ an element of $J$. I'm trying to think of an example in which $IJ$ does not equal $I \cap J$.
I'm thinking of letting $I = 2\mathbb{Z}$, and $J = \mathbb{Z}$, and $I\cap J = 2\mathbb{Z}$? Can someone point out anything faulty about this logic of working with an even ideal and then an odd ideal?
Thanks in advance.

But in your example $IJ = I = I \cap J$. How about taking $I = J = 2 \mathbb{Z}$. Then
$I \cap J = 2\mathbb{Z}$ while $IJ = 4\mathbb{Z}$. In general $IJ \subset I \cap J$, and the equality holds if $I+J = R$ where $R$ is the ring you are working with.
