Rings with a theory of greatest common divisors
type: Item
instance of: scholarly article
title: Rings with a theory of greatest common divisors
zbMATH Open document ID: 0901.13001
DOI: 10.1007/BF02678019
author: Friedemann Lucius
published in: Manuscripta Mathematica
publication date: 1998-05-13T00:00:00Z
full work available at URL: https://eudml.org/doc/156364
review text: We quote the author's words: ``In this paper we introduce the concept of a ring with gcd-theory to generalize the notion of a gcd-domain. We call a domain \(D\) ring with gcd-theory if there is a homomorphism \((\cdot)\) from \(D^*\) into a gcd-monoid \({\mathcal G}^+\) such that the divisibility properties of \(D\) are faithfully transferred to \({\mathcal G}^+\) and two elements of \({\mathcal G}^+\) are equal if they divide the same elements \((a)\), \(a\in D^*\). By means of *-ideal theory we study the fundamental properties of rings with gcd-theory. We show that a ring \(D\) with gcd-theory is simply a \(\nu\)-domain. We examine the behaviour of rings with gcd-theory when passing on to the integral closure in algebraic field extensions. Finally, we take a closer look at special cases of gcd-theories, namely to rings with gcd-theory of finite type, i.e. rings with gcd-theory in which any element of \({\mathcal G}^+\) can be written as the greatest common divisor of a finite number of elements \((a)\), \(a\in D^*\), and to the well-known rings with divisor theory''.
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F07
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F15
zbMATH DE Number: 1135229
zbMATH Keywords: \(\nu\)-domain
zbMATH Keywords: ring with gcd-theory
zbMATH Keywords: divisibility
zbMATH Keywords: *-ideal theory
zbMATH Keywords: integral closure
reviewed by: K. Lakkis
MaRDI profile type: Publication
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cites work: Q3994901
cites work: Quasi-divisor theories and generalizations of Krull domains
cites work: Q3137901
cites work: Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
cites work: Q5527106
cites work: Beiträge zur Arithmetik kommutativer Integritätsbereiche. II. \(v\)- Ideale und vollständig ganz abgeschlossene Integritätsbereiche
cites work: Q4344499
cites work: On Pruefer v-multiplication domains
cites work: Q4272883
cites work: Untersuchungen über Teilbarkeitseigenschaften in Körpern.
cites work: Idéaux divisoriels d'un anneau de polynômes
cites work: n-generator ideals in Prüfer domains
description: scientific article; zbMATH DE number 1135229