A Valuation Theorem for Noetherian Rings

Abstract

Let A⊂B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Denote by A‾ the integral closure of A in B. We show that A‾ is determined by finitely many unique discrete valuation rings. Our result generalizes Rees’ classical valuation theorem for ideals. We also obtain a variant of Zariski’s main theorem.