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:

(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.

(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

(2) $\text{depth} (M) + \text{depth}(N) = \dim A + \text{depth} {M\otimes_AN}$

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.

(3) Talking about intersection theory, by Serre formula for intersection multiplicity, 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.

Finally, if $V(I)$ and $V(J)$ only intersect at isolated closed points, (2) implies (1) locally on the support of the intersection, so

(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.

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.