A remark on Higman's result about separable algebras
type: Item
instance of: scholarly article
title: A remark on Higman's result about separable algebras
zbMATH Open document ID: 0603.16005
author: Koichiro Ohtake
author: Hiroshi Fukushima
published in: Tsukuba Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: According to a result due to \textit{D. G. Higman} [Can. J. Math. 7, 509-515 (1955; Zbl 0065.260)] a finite dimensional k-algebra A is separable iff there exist dual k-bases \(\{a_ i\}\) and \(\{b_ i\}\) such that \(\sum b_ ixa_ i\) is a unit for some \(x\in A\). The authors show that \(a_ i\) and \(b_ i\) can be rearranged here whereas the condition \(\sum a_ ixb_ i=1\) is not equivalent to the separability of A. The nondegeneracy of the trace form \(T_{A/k}\) is equivalent to the fact that \(\sum a_ ib_ i\) is a unit of A.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 3973072
zbMATH Keywords: finite dimensional k-algebra
zbMATH Keywords: dual k-bases
zbMATH Keywords: unit
zbMATH Keywords: separability
zbMATH Keywords: trace form
reviewed by: Feliks R. Bobovich
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160385
OpenAlex ID: W353381651
DOI: 10.21099/TKBJM/1496160385
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description: scientific article; zbMATH DE number 3973072