Intersections of symbolic powers of prime ideals

Let (R, m) be a local ring with prime ideals p and q such that rootp + a = m. If R is regular and contains a field, and dim(R/p) + dim(R/q) = dim(R), then it is proved that p((m)) boolean AND q((n)) subset of or equal to m(mdivided bym) for all positive integers m and n. This is proved using a generalization of Serre's Intersection Theorem which is applied 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) less than or equal to dim(R) holds when R is nonregular.