A valuation theorem for Noetherian rings

TL;DR

This paper generalizes valuation theorems for Noetherian rings, showing that the integral closure in finitely generated algebras is characterized by finitely many discrete valuation rings, extending classical results like Rees' and Zariski's theorems.

Contribution

It introduces a valuation theorem for Noetherian rings that broadens classical valuation results to a more general algebraic setting.

Findings

A' is determined by finitely many discrete valuation rings.

Generalization of Rees' classical valuation theorem.

A variant of Zariski's main theorem is obtained.

Abstract

Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. 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.