The recent work of Kurano and Roberts on Serre's positivity conjecture sugg
ests the following dimension inequality: for prime ideals p and q in a loca
l, Cohen-Macaulay ring (A, n) such that e(A(p)) = e(A) we have dim(A/p) +di
m(A/q) less than or equal to dim(A). We establish this dimension inequality
for excellent, local, Cohen-Macaulay rings which contain a field, for cert
ain low-dimensional cases and when R/p is regular.