Completions of Noetherian P.I. rings
type: Item
instance of: scholarly article
title: Completions of Noetherian P.I. rings
zbMATH Open document ID: 0704.16018
DOI: 10.1016/0021-8693(90)90273-Q
author: A. Braun
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let R be a left noetherian p.i. ring, and let M be a semimaximal ideal of R, the intersection of whose powers is zero. The author proves that \(\hat R=\lim_{\leftarrow i}R/M^ i\), the completion of R with respect to \(\{M^ i\}\), is left noetherian, thus answering a question that goes back to \textit{A. W. Goldie} [J. Algebra 5, 89-105 (1967; Zbl 0154.288)]. He also shows that the height of M may be different from the height of \(\hat M=\hat RM\hat R\). Furthermore, \(\hat R\) is not necessarily flat over R as it is in the commutative case. In fact, for a maximal ideal M, it is shown that \(\hat R\) is flat as a left R-module precisely when M is right localizable.
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P50
zbMATH DE Number: 4154666
zbMATH Keywords: left noetherian p.i. ring
zbMATH Keywords: semimaximal ideal
zbMATH Keywords: completion
zbMATH Keywords: height
zbMATH Keywords: flat
zbMATH Keywords: right localizable
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4154666