TITLE: When is the product of two ideals equal to their intersection?
QUESTION [54 upvotes]: Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds to the union $V(I\cap J)=V(I)\cup V(J)$. But a product $I.J$ gives a new subscheme $V(I.J)$ which has same support as the union but can be bigger in an infinitesimal sense. For example if $I=J$ you get a scheme $V(I^2)$ which is equal to "double" $V(I)$.
Vague Question  : What is geometric interpretation of $V(I.J)$ in general?
Precise question : When is $I\cap J=I.J$? Everybody knows the case $I+J=A$ but this is absolutely not necessary. For example if $A$ is UFD and $f,g$ are relatively prime then $(f).(g)=(f)\cap(g) $ but in general $(f)+(g)\neq A$ (e.g. $f=X, g=Y \in k[X, Y]$)
Thank you very much.

REPLY [12 votes]: I apologize in advance for resurrecting such an old question but I absolutely could not resist the urge of sharing a precise characterization for $I \cap J=IJ$, that I read recently in a beautiful short paper, when $I,J$ are monomial ideals in a polynomial ring.
First some terminology: Let $k$ be a field and $R=k[x_1,...,x_n]$ . Every monomial ideal $I$ of $R$ has a unique minimal monomial set of generators, usually denoted by $G(I)$ . For a set of monomials $T$ in $R$, let $\newcommand{\Supp}{\operatorname{Supp}}\Supp (T) :=\{i | x_i $ divides $m$ for some $m \in T \}$ .
With this, we can state the characterization: Let $I,J$ be monomial ideals in $k[x_1,...,x_n]$ , then $I \cap J=IJ$ if and only if $\Supp (G(I)) \cap \Supp (G(J))$ is empty.
This is Theorem 2.2 in https://link.springer.com/article/10.1007/s12044-019-0509-5
Here's the arxiv link https://arxiv.org/abs/1705.00488