Algebras having bases that consist solely of units
type: Item
instance of: scholarly article
title: Algebras having bases that consist solely of units
zbMATH Open document ID: 1355.16026
author: Sergio R. López-Permouth
author: Jeremy Moore
author: Nick Pilewski
author: Steve Szabo
published in: Israel Journal of Mathematics
publication date: 2015-11-16T00:00:00Z
review text: The authors study associative algebras \(A\) that contain a subalgebra \(R\) such that \(A\) is a free left \(R\)-module (they call these notions \(R\)-algebras) and there exists a basis \(B\) of \(A\) with the property that all its elements are invertible (an invertible basis). This paper extends and completes the result given in [the authors, Contemp. Math. 499, 219--228 (2009; Zbl 1191.16037)]. The authors classify this type of algebras with some additional properties over the invertible basis \(B\). For example they prove that an \(R\)-algebra \(A\) is a crossed product if and only if \(A\) contains an invertible basis \(B\) such that for every \(v\in B\) there exists \(\alpha\in U(R)\) with \(\alpha v^{-1}\in B\) and for every \(v\in B\) there exists \(\sigma_v\in \mathrm{Aut}(R)\) such that \(vr=\sigma_v(r)v\) for every \(r\in R\). Moreover, they prove that if \(A\) is an algebra over a ring \(R\) with a basis that includes \(1\), then \(M_n(A)\) is an (\(I2\)) algebra over \(R\), for \(n \geq 2\). Finally, they study finite-dimensional algebras over division rings and some invertible infinite-dimensional algebras.
Mathematics Subject Classification ID: 16S99
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 6509138
zbMATH Keywords: invertible algebras
zbMATH Keywords: crossed product
zbMATH Keywords: skew group ring
zbMATH Keywords: twisted group ring
zbMATH Keywords: group ring
MaRDI profile type: Publication
full work available at URL: s11856-015-1208-2
OpenAlex ID: W2269369957
cites work: The Leavitt path algebra of a graph.
cites work: Nonstable \(K\)-theory for graph algebras.
cites work: Q3655306
cites work: Bases consisting of units for Leavitt path algebras
cites work: Q3995022
cites work: Rings which are generated by their units
cites work: 2-GOOD RINGS
DOI: 10.1007/S11856-015-1208-2
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description: scientific article; zbMATH DE number 6509138