Going-down pairs of commutative rings
type: Item
instance of: scholarly article
title: Going-down pairs of commutative rings
zbMATH Open document ID: 1017.13004
DOI: 10.1007/BF02844429
author: R. Douglas Chatham
publication date: 2003-03-18T00:00:00Z
review text: An extension \(R\subseteq T\) of (commutative unitary) rings satisfies going-down (GD), if given prime ideals \(P_0\subset P\) in \(R\) and \(Q\) in \(T\) with \(Q\cap R=P\), there exists a prime ideal \(Q_0\) in \(T\) satisfying \(Q_0\subset Q\) and \(Q_0\cap R=P_0\). And a ring \(R\) is called GD-ring, if \(R/P\subset T\) satisfies GD for every prime ideal \(P\) of \(R\) and every domain \(T\) containing \(R/P\) [cf. \textit{D. E. Dobbs}, Houston J. Math. 23, 1-12 (1997; Zbl 0896.13006)]. In the paper under review, the author studies ring extensions \(R\subset T\), such that every intermediate ring extension \(R\subset A\subset B\subset T\) (respectively every intermediate ring \(R\subset A\subset T\)) satisfies GD (respectively is a GD-ring). For example, he shows that the paranthetical condition above holds for \(R\subset T\) with \(R\) zero-dimensional iff for each minimal prime ideal \(P\) of \(T\) the transcendence degree of \(T/P\) over \(R/P\cap R\) is \(\leq 1\). Related conditions, obtained replacing GD by a similar property are also studied, e.g. INC, open map spectra, finite fiber [INC means the ``incomparability property'' described by \textit{T. J. Papick}, Trans. Am. Math. Soc. 219, 1-37 (1976; Zbl 0345.13005); p. 28]. The paper contains numerous examples.
reviewed by: Tiberiu Dumitrescu
Mathematics Subject Classification ID: 13B24
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 1884331
zbMATH Keywords: going-down ring
zbMATH Keywords: zero-dimensional ring
MaRDI profile type: Publication
cites work: Remarks on generalized rings of quotients. IV
cites work: Q3488275
cites work: Krull Dimension in Power Series Rings
cites work: Pairs of Rings with the Same Prime Ideals
cites work: On seminormal overrings
cites work: Residually algebraic pairs of rings
cites work: Direct Products of Modules
cites work: The construction \(D+XD_s[X]\)
cites work: On Going Down For Simple Overrings II
cites work: Divided rings and going-down
cites work: Lying-Over Pairs of Commutative Rings
cites work: Going-down underrings
cites work: Prüfer's ascent result via Inc
cites work: Q4360752
cites work: Universally going-down homomorphisms of commutative rings
cites work: Integral Overrings of Two-Dimensional Going-Down Domains
cites work: Q3941522
cites work: Flat underrings
cites work: On Going-Down for Simple Overrings. III
cites work: Q4163618
cites work: Q4317713
cites work: A Generalization of Zariski's Main Theorem
cites work: Q5687544
cites work: Integral domains with Noetherian subrings
cites work: Q3137901
cites work: Q3953874
cites work: Q4049183
cites work: Topologically Defined Classes of Going-Down Domains
cites work: Generalized Quotient Rings
cites work: Pairs of Domains Where all Intermediate Domains are Noetherian
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1884331