CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS

Abstract

Let R be a (commutative integral) domain with quotient K; let R0 be the integral closure of R (in K). Then each overring of R (inside K) is a going-down domain if and only if R0 is a locally pseudo-valuation domain, T ⊆T0 satisfies going-down for every overring T of R, and tr. deg[VR0 (M)/M(R0)M :R0/M] ≤ 1 for every maximal ideal M of R0 (where VR0 (M) denotes the valuation domain that is canonically associated to the pseudo-valuation domain (R0)M). Additional equivalences are given in case R is locally finitedimensional. Applications include the case where R is integrally closed or R is not a Jaffard domain or R[X] is catenarian.