Azumaya-like properties of verbally prime algebras
type: Item
instance of: scholarly article
title: Azumaya-like properties of verbally prime algebras
zbMATH Open document ID: 0706.16014
DOI: 10.1016/0021-8693(90)90270-X
author: Allan Berele
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: This paper continues the investigations on the verbally prime algebras introduced by \textit{A. R. Kemer} [Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.5, 1042-1059 (1984; Zbl 0586.16010)]. Assume A is a verbally prime PI algebra over a field of characteristic 0 and Z is the centre of A. The main theorem in the paper states that there exists a non-nilpotent ideal M of Z satisfying the following conditions. 1. If \(c\in M\) then cA is contained in a finitely generated Z-module. 2. If \(J\triangleleft A\) then MJ\(\subset (J\cap Z)A\subset J\). 3. If \(J\triangleleft Z\) then M(JA\(\cap Z)\subset J\subset (JA\cap Z)\). 4. Assume \(\theta\) : \(A\otimes_ ZA^{op}\to End_ Z(A)\) is the canonical map. Then M Ker \(\theta\) \(=0\) and M End\({}_ Z(A)\subset Im \theta\). The proof uses the result quoted of Kemer and of \textit{Yu. P. Razmyslov} [Mat. Sb., Nov. Ser. 128, No.2, 194-215 (1985; Zbl 0601.16016)].
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 4158879
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: verbally prime algebras
zbMATH Keywords: prime PI algebra
zbMATH Keywords: centre
zbMATH Keywords: non- nilpotent ideal
zbMATH Keywords: finitely generated Z-module
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4158879