Projectively equivalent ideals and Rees valuations
type: Item
instance of: scholarly article
title: Projectively equivalent ideals and Rees valuations
zbMATH Open document ID: 1059.13001
author: Catalin Ciuperca
author: Louis J. Ratliff
author: William J. Heinzer
author: David E. Rush
published in: Journal of Algebra
publication date: 2004-11-29T00:00:00Z
review text: Rings considered are commutative, with at least two elements \(0\neq 1\). Projective equivalence of ideals was introduced by \textit{P. Samuel} [Ann. Math. (2) 56, 11--21 (1952; Zbl 0049.02301)] and his studies were continued by \textit{M. Nagata} [Mem. Coll. Sci., Univ. Kyoto, Ser. A 30, 165--175 (1957; Zbl 0095.02305)]. The authors build upon the work of \textit{S. McAdam, L. J. Ratliff jun}. and \textit{J. D. Sally} [in: Commutative algebra, Proc. Microprogram, Berkeley/CA (USA) 1989, Publ., Math. Sci. Res. Inst. 15, 391--405 (1989; Zbl 0742.13004)] who used some results of \textit{D. Rees} [J. Lond. Math. Soc. 31, 221--228 (1956; Zbl 0074.26303)]. Here is one of the authors' results: Let \(I\) and \(J\) be regular ideals of the Noetherian ring \(R\). The following are equivalent: (1) \(I\) and \(J\) are projectively equivalent, (2) The set of the Rees valuation rings for \(I\) and for \(J\) coincide and the values of \(I\) and \(J\) with respect to these Rees valuation rings are proportional. In particular, if the ideal \(I\) has only one Rees valuation ring \(V\), then the ideals \(J\) projectively equivalent to \(I\) are precisely the ideals \(J\) such that Rees\(J=\{V\}\). The strength of this paper is in its numerous examples and remarks.
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 2119558
zbMATH Keywords: projectively equivalent ideals
zbMATH Keywords: Rees valuations
zbMATH Keywords: Noetherian ring
zbMATH Keywords: integrally closed ideal
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.08.010
OpenAlex ID: W1993979436
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DOI: 10.1016/J.JALGEBRA.2004.08.010
description: scientific article; zbMATH DE number 2119558