Integrally closed ideals and type sequences in one-dimensional local rings
type: Item
instance of: scholarly article
title: Integrally closed ideals and type sequences in one-dimensional local rings
zbMATH Open document ID: 0926.13004
author: M. D'Anna
author: Donatella Delfino
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-11-24T00:00:00Z
full work available at URL: CONT27-4.html
review text: To any one-dimensional local noetherian analytically irreducible domain \(R\) one can associate a numerical semigroup \(v(R)\) by using the valuation \(v\) of the quotient field of \(R\). Let \(v(R) = \{0=s_0,s_1,\ldots,s_n,s_n+1,s_n+2,\ldots\}\), where \(s_n\) is the smallest integer such that all integers \(\geq s_n\) belong to \(v(R)\). Put \(I_j = \{x \in R\mid v(x)\geq s_j\}\) for \(j \leq n\). The sequence \(t_j(R) = \ell(I_j^{-1}/I_{j-1}^{-1})\) is called the type sequence of \(R\). The authors characterize the integrally closed ideals of \(R\) as the ideals of the form \(\{x \in R\mid v(x) \geq r\}\) for some \(r\) and give a criterion to check when the ideals \(I_j\) are stable. Moreover, they use the type sequence to study the difference \(t\ell(R/C)-\ell(\overline R/R)\), where \(t\) is the type of \(R\), \(\bar R\) is the integral closure of \(R\) in its quotient field and \(C\) denotes the conductor of \(R\) in \(\overline R\).
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 1206686
zbMATH Keywords: local noetherian analytically irreducible domain
zbMATH Keywords: valuation
zbMATH Keywords: numerical semigroup
zbMATH Keywords: type sequence
zbMATH Keywords: integrally closed ideal
reviewed by: Trung Ngo Viet
MaRDI profile type: Publication
OpenAlex ID: W2095342753
cites work: Q4292849
cites work: On the inequality \(\lambda (\bar R/R)\leq t(R)\lambda (R/{\mathfrak C})\) for one-dimensional local rings
cites work: Stable Ideals and Arf Rings
cites work: On the degree of singularity of one-dimensional analytically irreducible Noetherian local rings
DOI: 10.1216/RMJM/1181071860
description: scientific article; zbMATH DE number 1206686