Endomorphism rings of Leavitt path algebras.
type: Item
instance of: scholarly article
title: Endomorphism rings of Leavitt path algebras.
zbMATH Open document ID: 1331.16003
author: Gonzalo Aranda Pino
author: Kulumani M. Rangaswamy
author: Mercedes Siles Molina
published in: Journal of Pure and Applied Algebra
publication date: 2015-08-31T00:00:00Z
review text: Let \(L=L_K(E)\) be the Leavitt path algebra of the graph \(E\) and \(A=\text{End}(L_L)\) the endomorphism ring of \(L\). One of the first results of the paper states that \(L\) is a cyclic left \(A\)-module if and only if it is a cyclic right \(A\)-module, if and only if the graph has a finite number of vertices. Since \(L\) can be embedded in \(A\), it has sense to investigate how much room does \(L\) occupy within \(A\). The authors prove that \(A\) is an algebra of right quotients of \(L\). Furthermore the Jacobson radical of \(A\) is zero and \(A\) is non-singular. The paper contains also the result that \(A\) is an exchange ring if and only if \(L\) is an exchange ring if and only if \(E\) satisfies condition (K). Also \(L\) is a projective left \(A\)-module and a pure left submodule of \(A\). Section 3 starts investigating some regularity conditions which pass from \(A\) to \(L\) (\(\pi\)-regularity, von Neumann regularity, weak regularity). Then, assuming row-finiteness of the graph \(E\), it is proved the equivalence of the following: \(A\) is von Neumann regular. \(E\) is acyclic and every infinite path ends in a sink. \(L\) is both left and right self-injective and von Neumann regular. \(L\) is a semisimple right \(L\)-module. -- As a corollary \(\text{End}(L_L)\) is von Neumann regular if and only if \(\text{End}(_LL)\) is von Neumann regular. It is known that a module \(M\) is quasi-injective if and only if \(M\) is invariant under every endomorphism of its injective hull. A generalization of this condition leads to the concept of an automorphism invariant module. A right module \(M\) over a ring \(R\) is said to be automorphism invariant if \(M\) is invariant under every automorphism of its injective hull \(E(M)\). It is proved in the paper that a right module \(M\) over a ring \(R\) is both continuous and automorphism invariant if and only if \(M\) is quasi-injective. Then, for a row-finite graph \(E\) they have the equivalence of: \(L\) is automorphi…
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16S88
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16E50
zbMATH DE Number: 6476968
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: endomorphism rings
zbMATH Keywords: regularity conditions
zbMATH Keywords: self-injective modules
zbMATH Keywords: continuous modules
zbMATH Keywords: automorphism-invariant modules
zbMATH Keywords: exchange rings
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.05.016
OpenAlex ID: W2963508756
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DOI: 10.1016/J.JPAA.2015.05.016
description: scientific article; zbMATH DE number 6476968