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

Year 2014, Volume: 16 Issue: 16, 99 - 114, 01.12.2014

Ahmed Ayache David E. Dobbs

https://doi.org/10.24330/ieja.266230

Cited By: 2

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.

Keywords

Going-down domain, overring, treed domain, pseudo-valuation domain, valuation domain

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• Department of Mathematics University of Tennessee Knoxville, TN 37996-1320, U.S.A. e-mail: [e-mail]