Zassenhaus algebras.
type: Item
instance of: scholarly article
title: Zassenhaus algebras.
zbMATH Open document ID: 1245.16015
DOI: 10.1216/RMJ-2011-41-6-1793
author: Joshua Buckner
author: Manfred Dugas
published in: Rocky Mountain Journal of Mathematics
publication date: 2012-01-12T00:00:00Z
review text: If \(R\) is a ring and \(\mathcal F\) is a family of left ideals of \(R\), it is very useful to consider the subring \(\mathrm{End}(R,\mathcal F)\) of \(\mathrm{End}(R^+)\) which consists in all endomorphisms \(\varphi\) such that \(\varphi(X)\subseteq X\) for all \(X\in\mathcal F\). If \(\mathcal F\) consists of all left ideals of \(R\), then this subring is denoted by \(\widehat R\), and it consists of all endomorphisms of the Abelian group \(R^+\) which leave all left ideals invariant. These subrings are very useful in the constructions of modules (Abelian groups) with the endomorphism ring isomorphic to \(R\), [e.g. \textit{H. Zassenhaus} in J. Lond. Math. Soc. 42, 180-182 (1967; Zbl 0145.26405)]. The main aim of the present paper is to study \(\widehat R_K\) for the case \(R\) is a \(K\)-algebra over the field \(K\). There are many interesting results in this paper, and we will exemplify these with few examples. It is proved that if \(R\neq K\) is a finite dimensional local \(K\)-algebra and \(K\) is infinite then \(R\neq\widehat R_K\), i.e. \(R\) is not a Zassenhaus algebra, but \(\widehat{\widehat R}_K=\widehat R_K\), i.e. \(\widehat R_K\) is a Zassenhaus algebra (Theorem 3). If \(K\) is an infinite field and \(Q\) is a finite quiver, then the path algebra \(KQ\) is a Zassenhaus algebra. The same conclusion is valid for arbitrary fields \(K\) if we suppose that \(Q\) has not loops or it is acyclic (Theorem 5). It is also proved that the operator \(\widehat -\) acts in many important cases as a closure operator (Theorem 10).
reviewed by: Simion Sorin Breaz
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 5996725
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: Zassenhaus algebras
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: endomorphisms
zbMATH Keywords: endomorphism rings
zbMATH Keywords: families of left ideals
zbMATH Keywords: closure operators
MaRDI profile type: Publication
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full work available at URL: rmj-2011-41-6-1793
OpenAlex ID: W4244939238
description: scientific article; zbMATH DE number 5996725