Local-global results for modules over algebras and Azumaya rings
type: Item
instance of: scholarly article
title: Local-global results for modules over algebras and Azumaya rings
zbMATH Open document ID: 0742.16004
DOI: 10.1016/0021-8693(90)90300-D
author: Robert Wisbauer
published in: Journal of Algebra
publication date: 1992-06-25T00:00:00Z
review text: For any \(R\)-module \(M\), let \(\sigma[M]\) be the category of \(M\)-generated modules. The author calls a module \(M\) such that the functor \(\hbox{Hom}(M,-): \sigma[M]\to \hbox{End}(M)\hbox{-Mod}\) is an equivalence of categories a self-progenerator. Let \(Z\) be a commutative ring. A not necessarily associative algebra \(A\) can be viewed as a module over its multiplication algebra \(M(A)\). \(A\) is an Azumaya ring if it is a self-progenerator as \(M(A)\)-module, and an Azumaya algebra if it is a self-progenerator for all \(M(A)\)-modules. The author obtains local-global characterizations for Azumaya rings (Corollary 3.3), Azumaya algebras (Corollary 3.5) and biregular rings (Corollary 3.7, 3.8). Localization is taken with respect to maximal ideals of \(Z\) and with respect to the maximal ideals in the ring of idempotents of \(Z\) (Pierce stalks). The results include and improve corresponding results for associative rings by Burkholder, Dauns and Hofmann, Szeto and others.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 7134
zbMATH Keywords: equivalence of categories
zbMATH Keywords: self-progenerator
zbMATH Keywords: multiplication algebra
zbMATH Keywords: local-global characterizations
zbMATH Keywords: Azumaya rings
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: biregular rings
zbMATH Keywords: idempotents
zbMATH Keywords: Pierce stalks
reviewed by: M.-A. Knus
MaRDI profile type: Publication
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full work available at URL: 0021-8693(90)90300-d
OpenAlex ID: W1975370275
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description: scientific article; zbMATH DE number 7134