TITLE: Product of two ideals doesn't equal the intersection
QUESTION [12 upvotes]: 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.

REPLY [8 votes]: In a PID, we have $(a) \cap (b) = (\mathrm{lcm}(m,n))$, whereas we have $(a) \cdot (b) = (a \cdot b)$. So this ideal-theoretic question becomes a number-theoretic one and we see: $(a) \cap (b) = (a \cdot b)$ iff $a \cdot b$ is a lcm of $a$ and $b$ iff $a,b$ are coprime.