Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (I)

It is shown that the integral closure R' of a local (Noetherian) domain R is equal to the intersection of the Rees valuation rings of all proper ideals in R of the form (b, I_k)R, where b is an arbitrary nonzero nonunit in R and the I_k are an arbitrary descending sequence of ideals (varying with b and with I_k ⊆ (I_k-1 ∩ I₁^k) for all k > 1, one sequence for each b). Also, this continues to hold when b is restricted to being irreducible and no two distinct b are associates. We prove similar results for a Noetherian domain.