On a field-theoretic invariant for extensions of commutative rings, II

Abstract

This paper is a sequel. The earlier paper introduced, for any (unital) extension of (commutative unital) rings R T, an invariant L(T=R) defined as the supremum of the lengths of chains of intermediate fields in the extension kR(Q \ R) kT (Q), where Q runs over the prime ideals of T. Theorem 2.5 of that earlier paper calculated L(T=R) in case R T are (commutative integral) domains such that R T are “adjacent rings" (that is, in case R T is a minimal ring extension of domains). The statement of that Theorem 2.5 is incorrect for some adjacent rings R T such that R is integrally closed in T. Counterexamples are given to the original statement of Theorem 2.5. Two corrected versions of Theorem 2.5 are stated, proved and generalized from the domain-theoretic setting to the context of extensions of arbitrary rings. These results lead naturally to discussions involving the conductor (R : T) arising from a normal pair (R; T) of rings.