Finite normalizing extensions
type: Item
instance of: scholarly article
title: Finite normalizing extensions
zbMATH Open document ID: 0804.16025
DOI: 10.1016/0021-8693(92)90140-H
author: Ahmad Shamsuddin
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: Let the ring \(S\) be a finite normalizing extension of the ring \(R\). The author establishes a number of basic facts for this set-up, among which are descent of flatness (that is, if \((E\otimes_ R S)_ S\) is flat, then \(E_ R\) is flat) and that \(S\otimes_ RE=0\) implies that \(E=0\). The latter provides a very simple proof of properness, that is, \(SN\) is a proper left ideal of \(S\) whenever \(N\) is a proper left ideal of \(R\). The descent of projectivity is also established, and for the case when \(S_ R\) is flat, generalizations of these descents to homological dimensions are obtained. In particular, it is shown that if the left global dimension of \(R\) is finite, then it is bounded from above by that of \(S\).
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 130657
zbMATH Keywords: finite normalizing extension
zbMATH Keywords: descent of flatness
zbMATH Keywords: properness
zbMATH Keywords: proper left ideal
zbMATH Keywords: descent of projectivity
zbMATH Keywords: homological dimensions
zbMATH Keywords: left global dimension
MaRDI profile type: Publication
cites work: Q4023148
cites work: Q3786586
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Normalizing extensions and injective modules, essentially bounded normalizing extensions
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description: scientific article; zbMATH DE number 130657