Trace properties in rings with zero divisors
type: Item
instance of: scholarly article
title: Trace properties in rings with zero divisors
zbMATH Open document ID: 1248.13005
author: Thomas G. Lucas
author: Dawn McNair
published in: Journal of Algebra
publication date: 2012-02-11T00:00:00Z
review text: Let \(D\) be an integral domain with quotient field \(K\) and let for a nonzero ideal \(I\) of \(D,\) \(I^{-1}=D:I=\{x\in K:xI\subseteq D\}\). \textit{D. D. Anderson, J. A. Huckaba} and \textit{I. J. Papick} [Houston J. Math. 13, 13--17 (1987; Zbl 0624.13002)] showed that if \(D\) is a valuation domain then for every nonzero ideal \(I\) of \(D,\) either \(II^{-1}=D \) or \(II^{-1}\) is a prime ideal of \(D.\) Fontana, Huckaba and Papick, realizing \(II^{-1}\) as the trace of the \(D\)-module \(I\), introduced ``domains with trace property'' or TP-domains as domains \(D\) such that for each nonzero ideal \(I\) of \(D,\) \(II^{-1}=D\) or \(II^{-1}\) is a prime ideal of \(D.\) They looked for TP-domains among Noetherian and Prufer domains. Later, \textit{W. J. Heinzer} and \textit{I. J. Papick} [J. Algebra 112, No. 1, 110--121 (1988; Zbl 0641.13001)] extended the playing field \ by defining domains with the radical trace property (RTP) as domains \(D\) such that for each nonzero non-invertible ideal \(I\) of \(D\) the ideal \(II^{-1}\) is a radical ideal\(.\) A number of algebraists, including the first author, S. Kabbaj and A. Mimouni have worked on topics related to this RTP-property. The ideal \(II^{-1}=I(D:I)\) is also known as the trace ideal. Often when a topic has been sufficiently explored for integral domains some researchers turn their attention to seeing if similar results can be obtained for rings with zero-divisors. I tend to think that this trend started, in earnest, in the sixties and grew with \textit{J. Huckaba} [Commutative rings with zero divisors. New York etc.: Marcel Dekker, Inc. (1988; Zbl 0637.13001)], who was the advisor of the first author of the paper under review. In any case it did produce some interesting results and the paper under view is one of them. So, from now on, let \(R\) be a commutative ring with \(1\neq 0\), \(T(R)\) the total quotient ring of \(R,\) \(Q(R)\) the complete ring of quotients and let \(Q_{0}(R)\) be the ring of fin…
reviewed by: Muhammad Zafrullah
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 6005466
zbMATH Keywords: radical trace property
zbMATH Keywords: trace ideal
zbMATH Keywords: ring of finite fractions
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.05.039
OpenAlex ID: W2080980748
cites work: Remarks on generalized rings of quotients
cites work: Remarks on generalized rings of quotients. III
cites work: Q3762418
cites work: Q3041324
cites work: On the ubiquity of Gorenstein rings
cites work: Annihilator ideals, associated primes and kasch-mccoy commutative rings
cites work: Domains satisfying the trace property
cites work: Prüfer domains and endomorphism rings of their ideals
cites work: Q3137901
cites work: Valuations and Prufer Rings
cites work: The radical trace property
cites work: When is the dual of an ideal a ring?
cites work: Q4039904
cites work: When the dual of an ideal is a ring
cites work: Q4517583
cites work: Manis valuations and Prüfer extensions. I: A new chapter in commutative algebra
cites work: Q5647004
cites work: Autour de la platitude
cites work: Strong Prüfer rings and the ring of finite fractions
cites work: The radical trace property and primary ideals
cites work: Krull rings, Prüfer \(v\)-multiplication rings and the ring of finite fractions
cites work: Q4343557
cites work: Trace Properties and Integral Domains, II
cites work: Q5392459
cites work: Sur la compacité du spectre minimal d'un anneau
DOI: 10.1016/J.JALGEBRA.2011.05.039
description: scientific article; zbMATH DE number 6005466