Azumaya rings with locally perfect centers
type: Item
instance of: scholarly article
title: Azumaya rings with locally perfect centers
zbMATH Open document ID: 0647.16005
DOI: 10.1016/0021-8693(86)90155-9
author: Douglas G. Burkholder
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: An Azumaya ring is a ring R in which i) every ideal is generated by central elements, ii) every ring homomorphism from R maps the centre of R onto the centre of the image of R. If R is an Azumaya ring with centre Z then R/RM is a central simple Z/M-algebra for every maximal ideal M of Z. For the converse, the classical known results show that this is true for rings which are module finite over their centre. In this paper the author exhibits that the converse is true whenever Z is locally perfect. The author sharpens the condition on T by showing that if T is a locally perfect subring of the centre of R, such that R/RM is a central simple T/M-algebra for every maximal ideal M of T, then R is an Azumaya ring with centre T. This allows the author to exhibit that the central ideal algebras over perfect rings due to \textit{M. L. Ranga Rao} [Bull. Am. Math. Soc. 78, 588-592 (1972; Zbl 0258.16004)] are indeed Azumaya rings.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16L30
zbMATH DE Number: 4055803
zbMATH Keywords: central elements
zbMATH Keywords: centre
zbMATH Keywords: central simple Z/M-algebra
zbMATH Keywords: locally perfect subring
zbMATH Keywords: maximal ideal
zbMATH Keywords: central ideal algebras
zbMATH Keywords: perfect rings
zbMATH Keywords: Azumaya rings
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90155-9
OpenAlex ID: W2059440897
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description: scientific article; zbMATH DE number 4055803