Normal Pairs of Going-down Rings

• Journal title : Kyungpook mathematical journal
• Volume 51, Issue 1,  2011, pp.1-10
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2011.51.1.001
Title & Authors
Normal Pairs of Going-down Rings
Dobbs, David Earl; Shapiro, Jay Allen;

Abstract
Let (R, T) be a normal pair of commutative rings (i.e., R $\small{{\subseteq}}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R $\small{{\subseteq}}$ S $\small{{\subseteq}}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.
Keywords
Normal pair;prime ideal;total quotient ring;valuation domain;divided domain;pullback;going-down ring;EGD ring;locally divided ring;weak Baer ring;reduced ring;
Language
English
Cited by
1.
Characterizing the ring extensions that satisfy FIP or FCP, Journal of Algebra, 2012, 371, 391
2.
On Finite Maximal Chains of Weak Baer Going-Down Rings, Communications in Algebra, 2012, 40, 5, 1843
3.
A GENERALIZATION OF PRÜFER'S ASCENT RESULT TO NORMAL PAIRS OF COMPLEMENTED RINGS, Journal of Algebra and Its Applications, 2011, 10, 06, 1351
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