An additivity principle for p.i. rings
type: Item
instance of: scholarly article
title: An additivity principle for p.i. rings
zbMATH Open document ID: 0576.16018
DOI: 10.1016/0021-8693(85)90020-1
author: A. Braun
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let \({\mathcal R}\) be a prime ring with polynomial identity, \({\mathcal Z}\) is the center of \({\mathcal R}\) and \({\mathcal T}({\mathcal R})\) is the trace ring which is generated by the coefficients of the characteristic polynomials of elements of \({\mathcal R}\) over \({\mathcal Z}\). Let also \(p\in Spec {\mathcal Z}\), \({\mathcal I}=\{{\mathcal Q}\); \({\mathcal Q}\in Spec {\mathcal R}\), \({\mathcal Q}\cap {\mathcal Z}=p\}\), \(| {\mathcal I}| =s\). The author proves that if \({\mathcal T}({\mathcal R})={\mathcal R}\) then p.i.d.(\({\mathcal R})=\sum^{s}_{i=1}K_ i\cdot (p.i.d.({\mathcal R}/{\mathcal P}_ i))\), where \({\mathcal P}_ i\in {\mathcal I}\), \(k_ i\geq [{\mathcal Z}({\mathcal R}_ p/{\mathcal P}_{ip})_{sep}:{\mathcal Z}_ p/{\mathcal P}_ p]\), \(i=1,2,...,s\). As a corollary the author gives a new proof of the famous Bergman-Small theorem.
Mathematics Subject Classification ID: 16Rxx
zbMATH DE Number: 3920713
zbMATH Keywords: prime ring with polynomial identity
zbMATH Keywords: center
zbMATH Keywords: trace ring
zbMATH Keywords: Bergman-Small theorem
reviewed by: Yu. N. Maltsev
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90020-1
OpenAlex ID: W2066893743
cites work: On the characteristic polynomial of a sum of matrices
cites work: Prime ideals in PI-rings
cites work: Integral ring homomorphisms
cites work: P.I. degrees and prime ideals
cites work: Affine polynomial identity rings and their generalizations
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description: scientific article; zbMATH DE number 3920713