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.



When is the product of two ideals equal to their intersection?
To add to David Speyer's answer, since this story continues with a rather interesting and illustrious history:


When $A$ is regular, the Tor functor satisfies the following property:



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> (1) $\text{Tor}\_1^A(M,N) = 0$ implies $\text{Tor}\_i^A(M,N) = 0$ for $i>0$ for any two finitely generated modules.
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(this is a theorem by Auslander in the geometric and unramified case and Lichtenbaum in the ramified case. (1) is called the *rigidity* of Tor).


It turns out that when $A$ is regular and local (so one can talk about depth), (1) implies



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> (2) $\text{depth} (M) + \text{depth}(N) = \dim A + \text{depth} {M\otimes\_AN}$
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This stunning formula looks exactly the same as the property of "proper intersection" in intersection theory, except that one uses depth instead of dimension. Note that if $M=A/I, N=A/J$ then $M\otimes N = A/(I+J)$, which represents the intersection of $V(I)$ and $V(J)$, so this is very geometric.



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> (3) Talking about intersection theory, by [Serre formula for intersection multiplicity](https://en.wikipedia.org/wiki/Serre_multiplicity_conjectures), as all the Tors vanish, one can compute the intersection multiplicity of $V(I), V(J)$ by counting the length at the minimal components (i.e. the naive way). So you will have a generalization of Bezout theorem.
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Finally, if $V(I)$ and $V(J)$ only intersect at isolated closed points, (2) implies (1) locally on the support of the intersection, so



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> (4) If $V(I) \cap V(J)= \{m\_1, \cdots, m\_n \}$ then $I\cap J = IJ$ if and only if $A/I, A/J$ are locally Cohen-Macaulay at the points $m\_i$s.
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You can find the last statement in Serre's Local Algebra book, V.6, Theorem 4, p 110 of the English version.


PS: Also, David did not mention his own interesting contribution, [here](https://arxiv.org/abs/math/0601202).


