Applications of the Theory of Valuation Rings

Abstract

The theory of valuation rings was founded in connection with number theory. The valuations concerned are multiplicative valuations having values in non-negative real numbers. This chapter describes the notion of an additive valuation (called an “Allgemeine Bewertung”) having values in an ordered additive group. This is a generalization of a p-adic valuation that Krull employed to study commutative rings. The chapter reviews several well known and important applications of the theory of valuations. It presents the proof of a generalization of a well known existence theorem of valuation rings. The chapter discusses a theorem that provides a sufficient condition for a field to be finitely generated, as an application of the theory of valuation rings. Several modifications of the theorem are also presented. The notion of Krull ring is a generalization of that of noetherian normal ring and plays an important role in studying noetherian rings.