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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 satisfying 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 $R/I, R/J$ A/I, A/J$are locally Cohen-Macaulay at the points$m_i$s. You can find the last statement in Serre's Local Algebra book, V.6, Theorem 4, p 110 of the English version. PS: Also, David did not mention his own interesting contribution, here. 1 To add to David Speyer's answer, since this story continues with a rather interesting and illustrious history: When$A$is regular, the Tor functor satisfying the following property: (1)$\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$R/I, R/J$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.