Leavitt path algebras having unbounded generating number
type: Item
instance of: scholarly article
title: Leavitt path algebras having unbounded generating number
zbMATH Open document ID: 1355.16025
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Journal of Pure and Applied Algebra
publication date: 2017-01-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1603.09695
review text: The authors consider the following sequence of ring theoretic properties: {\parindent=0.7cm\begin{itemize}\item[(1)] Invariant basis number (IBN)\item[(2)] Unbounded generating number (UGN), that is, for each positive integer \(m\), any set of generators for the free right \(R\)-module \(R^m\) has cardinality \(\geq m\). \item[(3)] Stable finiteness, that is, for any \(n\in\mathbb N^+\), \(R^n\cong R^n\oplus K\) (as right R-modules) implies \(K=0\). \item[(4)] Hermite property, that is, for all \(m,n\in\mathbb N\) and any right \(R\)-module \(K\), \(R^n\cong R^m\oplus K\) (as right \(R\)-modules) implies that \(n\geq m\) and \(K=R\). \item[(5)] Cancellation of projectives: for any finitely generated projective right \(R\)-modules \(P\) and \(P'\), \(P\oplus R\cong P'\oplus R\) (as right \(R\)-modules) implies that \(P\cong P'\). \end{itemize}} The paper under review studies some of the notions above in the ambient space of Leavitt path algebras (\(L(E)\) in the sequel) and also in that of Cohn path algebras (denoted \(C(E)\) in the sequel). Recently, the first author and \textit{M. Kanuni} [Commun. Algebra 44, No. 1, 371--380 (2016; Zbl 1354.16032)] have shown that \(C(E)\) has IBN for every finite graph \(E\). The authors point out that, at of the writing of this article, it is an open question to give necessary and sufficient conditions on \(E\) so that \(L(E)\) has IBN. However, regarding the remaining four aforementioned properties, they completely classify those graphs \(E\) for which \(L(E)\) has UGN (Theorem 3.16), as well as classify those graphs \(E\) for which \(L(E)\) satisfies (equivalently) properties (3), (4), and (5) (Theorem 4.2). They achieve similar results for \(C(E)\) as well.
Mathematics Subject Classification ID: 16S99
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6676879
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: unbounded generating number
zbMATH Keywords: Morita invariant
zbMATH Keywords: unital rings
reviewed by: Candido Martín González
MaRDI profile type: Publication
OpenAlex ID: W2964206655
arXiv ID: 1603.09695
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DOI: 10.1016/J.JPAA.2016.09.014
description: scientific article