  Let (R,m) be a local ring with prime ideals p and q such that p+q is an
m-primary ideal. If R is regular and contains a field, and
dim(R/p)+dim(R/q)=dim(R), we prove that p^{(r)}\cap q^{(n)}\subseteq m^{m+n}
for all positive integers r and s. This is proved using a generalization of
Serre's Intersection Theorem which we apply to a hypersurface R/fR. The
generalization gives conditions that guarantee that Serre's bound on the
intersection dimension dim(R/p)+dim(R/q) \leq dim(R) holds when R is
nonregular.

Commutative Algebra