Domains satisfying the trace property
type: Item
instance of: scholarly article
title: Domains satisfying the trace property
zbMATH Open document ID: 0612.13010
DOI: 10.1016/0021-8693(87)90083-4
author: Marco Fontana
author: James A. Huckaba
author: Ira Papick
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let R be a commutative integral domain with identity. R is said to satisfy the trace property if for each R-module M, the trace ideal of M is either equal to R or is a prime ideal. It is shown that R satisfies the trace property if and only if for each nonzero ideal I of R, either \(II^{-1}=R\) or is a prime ideal. It was shown by the reviewer and \textit{J. A. Huckaba} and \textit{I. J. Papick} [Houston J. Math. 13, 13-17 (1987)], that this last condition holds for a valuation domain. The purpose of this paper is to characterize certain classes of domains satisfying the trace property. It is shown that a Noetherian domain R satisfies the trace property if and only if R is Dedekind or \(\dim (R)=1\), R has a unique non-invertible maximal ideal M and \(M^{-1}\) is equal to the integral closure of R. The question of when a Prüfer domain satisfies the trace property is also considered.
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 3989505
zbMATH Keywords: trace property
zbMATH Keywords: Noetherian domain
zbMATH Keywords: Prüfer domain
reviewed by: D. D. Anderson
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90083-4
OpenAlex ID: W2028982206
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description: scientific article; zbMATH DE number 3989505