Krull rings, Prüfer \(v\)-multiplication rings and the ring of finite fractions
type: Item
instance of: scholarly article
title: Krull rings, Prüfer \(v\)-multiplication rings and the ring of finite fractions
zbMATH Open document ID: 1102.13021
author: Thomas G. Lucas
published in: Rocky Mountain Journal of Mathematics
publication date: 2006-01-24T00:00:00Z
review text: This paper deals with extending the notions of Krull domains and PvMDs to rings with zero divisors. Two of the problems to address involve characterizing when a Nagata ring will be a Krull ring and when it will be a PvMR. For both problems, the characterizations entail considering how the ring in question is placed in its associated ring of finite fractions. Reviewer's remark. The notion of the ring of finite fractions was introduced by the author in previous papers [J. Pure Appl. Algebra 61, 49--52 (1989; Zbl 0697.13003) and in: Commutative ring theory (Fès, 1992), Lecture Notes Pure Appl. Math. 153, 181--191 (1994; Zbl 0865.13003)]. The author found some interesting applications of this ring, which has a very significant role in the paper under review. For these reasons, and also because the reviewer finds the ``finite fractions'' interesting and also rather curious, the definition of the ring of finite fractions is given here. Let \(R\) be a commutative ring with identity. As usual, we use \(R[X]\) to denote the polynomial ring over \(R\). We define the ring of finite fractions over \(R\) as the subring of the total quotient ring of \(R[X]\) consisting of the following fractions: \(f=b(X)/a(X)\), where \(a(X)=\sum a_iZ^i\), \(b(X)=\sum b_jX^j\in R[X]\), with \(\deg(b(X))\leq \deg(a(X))\) such that \(fa_i=b_i\) for each coefficient \(a_i\) of \(a(X)\).
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 5001008
zbMATH Keywords: PvMR rings
zbMATH Keywords: finite fractions
zbMATH Keywords: Krull domains
zbMATH Keywords: Nagata ring
reviewed by: Ezio Stagnaro
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069687
OpenAlex ID: W2089217833
cites work: Q3898597
cites work: On the normality of R(X)
cites work: Essential valuations of krull rings with zero divisiors
cites work: The rings R(X) and \(R<X>\)
cites work: On the ubiquity of Gorenstein rings
cites work: Ideals in rings of integer valued polynomials.
cites work: Integer-valued polynomials
cites work: Les idéaux premiers de l'anneau des polynomes à valeurs entières.
cites work: Q5670698
cites work: Q3137901
cites work: Semigroup rings as Prüfer rings
cites work: Some Results on v-Multiplication Rings
cites work: Prüfer rings with zero divisors.
cites work: Some remarks on star-operations
cites work: The generalized Kronecker function ring and the ring R(X).
cites work: Characterizations of *-Multiplication Domains
cites work: On t-invertibility II
cites work: Q4039904
cites work: Annihilation of ideals in commutative rings
cites work: Quotient rings of polynomial rings
cites work: On the converse of a well-known fact about Krull domains
cites work: A characterization of Krull rings with zero divisors
cites work: Characterizations of Krull rings with zero divisors
cites work: A Quasi-Commutative Ring That is Not Neo-Commutative
cites work: Krull rings
cites work: Q3760508
cites work: Characterizing When R[ X ] is Integrally Closed
cites work: Characterizing when R[X] is integrally closed. II
cites work: The Complete Integral Closure of R[ X ]
cites work: Strong Prüfer rings and the ring of finite fractions
cites work: The integral closure of R(X) and R(X)
cites work: Q4343557
cites work: On t-invertibility
cites work: Valuations on a Commutative Ring
cites work: Q3967638
cites work: Notes on PrÜfer v-Multiplication Rings
cites work: Q5735274
cites work: Q5682087
cites work: Q3943055
cites work: Krull rings with zero divisors
cites work: Sur la compacité du spectre minimal d'un anneau
cites work: Some polynomial characterizations of Prüfer v-multiplication domains
DOI: 10.1216/RMJM/1181069687
Recommended article: Q3653951
Recommended article: Equations for the set of overrings of normal rings and related ring extensions
Recommended article: The rings R(X) and \(R<X>\)
Recommended article: Q3728115
Recommended article: Every divisor class of Krull monoid domains contains a prime ideal
Recommended article: Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
Recommended article: THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS
Recommended article: A characterization of Krull rings with zero divisors
Recommended article: Q3142917
Recommended article: On v-domains: a survey
description: scientific article; zbMATH DE number 5001008