On universally going-down underrings. II
type: Item
instance of: scholarly article
title: On universally going-down underrings. II
zbMATH Open document ID: 0679.13004
DOI: 10.1007/BF01189131
author: David E. Dobbs
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: [For part I see ibid. (to appear; Zbl 0665.13006).] This paper gives a new characterization of an often-characterized class of integral domains. Let R be an integral domain with quotient field K. It is proved that the inclusion map \(A\to B\) satisfies universally going down [in the sense of the author and \textit{M. Fontana}, J. Algebra 90, 410-429 (1984; Zbl 0544.13004)] for each pair \(A\subset B\) of subrings of R having quotient field K if and only if either R is isomorphic to an overring of \({\mathbb{Z}}\) or \(R=K\) is an algebraic field extension of a finite field. This result sharpens a theorem of part I of this paper. The proof uses a result of the author about the analogous situation involving going-down [from Bull. Aust. Math. Soc. 36, 503-513 (1987; Zbl 0619.13004)], as well as standard facts about weak/semi-normalization and pullbacks.
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 4112732
zbMATH Keywords: integral domains
zbMATH Keywords: universally going down
zbMATH Keywords: semi-normalization
MaRDI profile type: Publication
cites work: Q3349016
cites work: Q5576163
cites work: On Going Down For Simple Overrings II
cites work: Going-down underrings
cites work: On universally going-down underrings
cites work: Universally going-down homomorphisms of commutative rings
cites work: Universally going-down integral domains
cites work: Topologically defined classes of commutative rings
cites work: On seminormality
cites work: Q3249822
full work available at URL: https://doi.org/10.1007/bf01189131
OpenAlex ID: W4256683501
description: scientific article; zbMATH DE number 4112732