On a Field-Theoretic Invariant for Extensions of Commutative Rings#

Abstract

If R ⊆ T is an extension of (commutative integral) domains, Λ(T/R) is defined as the supremum of the lengths of chains of intermediate fields in the extension k R (Q ∩ R) ⊆ k T (Q), where Q runs over the prime ideals of T. The invariant Λ(T/R) is determined in case R and T are adjacent rings and in case Spec(R) = Spec(T) as sets. It is proved that if R is a domain with integral closure R′, then Λ(T/R) = 0 for all overrings T of R if and only if R′ is a Prüfer domain such that Λ(R′/R) = 0. If R ⊆ T are domains such that the canonical map Spec(T) → Spec(R) is a homeomorphism (in the Zariski topology), then Λ(T/R) is bounded above by the supremum of the lengths of chains of rings intermediate between R and T. Examples are given to illustrate the sharpness of the results.