Strongly bounded rings
type: Item
instance of: scholarly article
title: Strongly bounded rings
zbMATH Open document ID: 0926.13002
author: Moira A. McDermott
published in: Journal of Algebra
publication date: 1999-11-09T00:00:00Z
review text: The main result of this paper is the following condition for rings in which tight closure commutes with localization. Let \(R\) be a commutative noetherian ring of characteristic \(p\). Then \(R\) should have a test element \(c\) such that there exists an \(R\)-linear map \(\theta: R^{1/q} \to R^{1/pq}\) taking \(c^{1/q}\) to \(c^{1/pq}\) for some power \(q= p^e\). This condition also gives an effective bound for computing tight closure. The author terms such a ring \(R\) strongly bounded with respect to \(c\). She goes on to show that any \(F\)-finite domain \(R\) whose integral closure \(S\) is regular is strongly bounded with respect to an element \(c \in R \setminus \{0\}\) such that \(cS \subseteq R\). In particular, this includes the case of one-dimensional \(F\)-finite domains.
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13B22
zbMATH DE Number: 1117788
zbMATH Keywords: tight closure
zbMATH Keywords: localization
zbMATH Keywords: Frobenius endomorphism
zbMATH Keywords: integral closure
zbMATH Keywords: strongly bounded ring
reviewed by: Trung Ngo Viet
MaRDI profile type: Publication
OpenAlex ID: W2077739977
cites work: Localization of tight closure and modules of finite phantom projective dimension.
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: Tight closure and elements of small order in integral extensions
cites work: F-Regularity, Test Elements, and Smooth Base Change
cites work: Q4871734
cites work: Tight closure, plus closure and Frobenius closure in cubical cones
DOI: 10.1006/JABR.1997.7208
description: scientific article; zbMATH DE number 1117788