Abstract
This chapter reviews derivations and valuation rings. The derivations are related to contact, and so are valuations, so one may ask for a study connecting derivations and valuations. . The chapter describes an existence theorem: let 0 = k[x1…, xn] be a finite integral domain over a base field k of characteristic zero, let m be a prime ideal therein, and let D be an integral derivation of the local ring 0m {i.e., D0m ⊂ 0m). Then, there exists a valuation ring centered at m that is also sent into itself by D.. The chapter also highlights that if a derivation of a local ring of an algebraic variety sends the ring into itself, then it also sends some dominating valuation ring into itself. If the ring is 2-dimensional and regular and the maximal ideal is not differential, then the valuation ring is unique. The ground field is assumed to be of characteristic 0.
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