Linear growth of primary decompositions of integral closures
type: Item
instance of: scholarly article
title: Linear growth of primary decompositions of integral closures
zbMATH Open document ID: 0932.13003
author: R. Y. Sharp
published in: Journal of Algebra
publication date: 1998-12-13T00:00:00Z
review text: Let \(R\) be an commutative noetherian ring and \(I\) a proper ideal of \(R\). \textit{I. Swanson} has shown [Math. Ann. 307, No. 2, 299-313 (1997; Zbl 0869.13001)] that \(I\) has linear growth of primary decompositions, that is there exists a positive integer \(h\) such that, for every positive integer \(n\), the ideal \(I^n\) has a minimal primary decompositin \(I^n=Q_{n1}\cap \dots\cap Q_{nk_n}\) such that \((\sqrt{Q_{ni}})^{hn}\subseteq Q_{ni}\) for all \(i=1,\dots,k_n\). In the paper under review, the author establishes a parallel result for integral closures of ideals, that is, there exists a positive integer \(h\) such that, for every positive integer \(n\), the integral closure \(\overline{I^n}\) of \(I^n\) has a minimal primary decomposition \(\overline{I^n}=Q_{n1}\cap \dots\cap Q_{nk_n}\) such that \((\sqrt{Q_{ni}})^{hn}\subseteq Q_{ni}\) for all \(i=1,\dots,k_n\). His proof uses the extended Rees algebra and some properties of the Nagata rings to reduce the assertion to the case in which \(I\) is principal generated by a non-zerodivisor and \(R\) is a reduced normal Nagata ring.
reviewed by: Tiberiu Dumitrescu
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13F99
zbMATH DE Number: 1213856
zbMATH Keywords: primary decomposition
zbMATH Keywords: noetherian ring
zbMATH Keywords: Rees algebra
zbMATH Keywords: Nagata ring
zbMATH Keywords: integral closures of ideals
MaRDI profile type: Publication
full work available at URL: jabr.1998.7436
OpenAlex ID: W2046727517
DOI: 10.1006/JABR.1998.7436
Recommended article: LINEAR GROWTH OF PRIMARY DECOMPOSITIONS OF MODULES AND INTEGRAL CLOSURES
Recommended article: Growth of primary decompositions of Frobenius powers of ideals
Recommended article: Primary decomposition: Compatibility, independence and linear growth
Recommended article: Primary decomposition. II: Primary components and linear growth
Recommended article: Ratliff–Rush closures and linear growth of primary decompositions of ideals
Recommended article: Powers of ideals. Primary decompositions, Artin-Rees lemma and regularity
Recommended article: A generalized principal ideal theorem
Recommended article: Clôture Intégrale D'idéaux et la Propriété (Zk)
Recommended article: Integral Closures of Noetherian Integral Domains as Intersections
Recommended article: Commutative ideal theory without finiteness conditions: Primal ideals
description: scientific article; zbMATH DE number 1213856