Multi-ideal-adic completions of Noetherian rings
type: Item
instance of: scholarly article
title: Multi-ideal-adic completions of Noetherian rings
zbMATH Open document ID: 1184.13027
author: C. Rotthaus
author: Sylvia Wiegand
author: William J. Heinzer
published in: Michigan Mathematical Journal
publication date: 2009-09-17T00:00:00Z
review text: Let \(R\) be a commutative ring with \(1\) and let \(\{I_{n}\}_{n=0}^{\infty }\) be a filtration of ideals of \(R\), i.e., \(I_{0}\supseteq I_{1}\supseteq I_{2}\supseteq \dots\). We can make \(R\) a topological space w.r.t. the filtration \(\{I_{n}\}\) by taking the cosets \(r+I_{n},\) \(r\in R\) and \(n\in Z_{+},\) as basic open sets. \(R\) is Hausdorff under this filtration topology if and only if \(\bigcap I_{n}=(0).\) Call \(R\) complete with respect to a filtration \(\{I_{n}\}\) if (a) \(\bigcap I_{n}=(0)\) and (b) every Cauchy sequence in \(R\) converges to an element of \(R.\) In ring theory, the completion of a ring \(R\) w.r.t. a filtration \(\{I_{n}\}\) is a complete ring \(R^{\ast }= \underleftarrow{\lim_{n}}\{R/I_{n}\}\) with a well-defined filtration, and a canonical homomorphism \(\psi :R\rightarrow R^{\ast },\) see chapter 9 of \textit{D. G. Northcott} [Lessons on rings, modules and multiplicities. Reprint of the 1968 hardback ed. Cambridge: Cambridge University Press (2008; Zbl 1185.13001)]. The canonical homomorphism \(\psi \) is injective, rendering ring theoretic completion closest to topological completion, precisely when \( \bigcap I_{n}=(0)\). A filtration \(\{I_{n}\}_{n=0}^{\infty }\) is called multiplicative if \( I_{0}=R \) and \(I_{m}I_{n}\subseteq I_{m+n}.\) A well-known multiplicative filtration on \(R\) is the \(I\)-adic filtration \(\{I^{n}\}_{n=0}^{\infty }\), where \(I\) is a fixed (proper) ideal of \(R.\) Now if \(R\) is noetherian with nonzero Jacobcon radical \(J\), as in the paper under review, then \(\bigcap _{n=0}^{\infty }J^{n}=(0).\) The authors introduce a filtration \( \{I_{n}\}_{n=0}^{\infty }\) requiring that \(I_{n}\subseteq J^{n}\) for each \( n>0\) and that \(I_{nk}\subseteq I_{n}^{k}\) for all \(n,k\geq 0.\) They call the completion resulting from this, somewhat restricted, filtration a multi-adic completion. One of the motivations for this study is that several useful examples can be considered as subrings of \(R^{\as…
reviewed by: Muhammad Zafrullah
Mathematics Subject Classification ID: 13B35
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13F40
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13J15
zbMATH DE Number: 5604539
zbMATH Keywords: filtration
zbMATH Keywords: completion
zbMATH Keywords: I-adic
zbMATH Keywords: multi-adic completion
zbMATH Keywords: excellent
zbMATH Keywords: Henselian
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1220879416
OpenAlex ID: W2033562688
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DOI: 10.1307/MMJ/1220879416
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description: scientific article; zbMATH DE number 5604539