Integral Extensions

Abstract

A ring morphism φ:R⟶R′ is called integral, or an integral extension, if every element x∈R′ satisfies an integral equation over R, i.e. an equation of type x ⁿ+a ₁ x ⁿ⁻¹+…+a _n =0 for suitable coefficients a _i ∈R where R′ is viewed as an R-module via φ. The first objective of the chapter is to characterize integral extensions in terms of finite extensions and to show that finite extensions are integral. Together with Noether’s Normalization Lemma this fundamental fact leads to the characterization of finitely generated algebras over fields and, in particular, to Hilbert’s Nullstellensatz. Finally, the behavior of chains of prime ideals with respect to integral extensions is studied by means of the Cohen–Seidenberg Theorems.