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GRADED CHAIN CONDITIONS AND LEAVITT PATH ALGEBRAS OF NO-EXIT GRAPHS

Part of: Rings and algebras with additional structure Rings and algebras arising under various constructions Modules, bimodules and ideals

Published online by Cambridge University Press:  12 December 2017

LIA VAŠ
LIA VAŠ*
Affiliation:
Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA 19104, USA email l.vas@usciences.edu
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Abstract

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We obtain a complete structural characterization of Cohn–Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancelative. We introduce the nonunital generalizations of graded analogs of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams–Aranda–Perera–Siles characterization of locally noetherian and locally artinian Leavitt path algebras.

Keywords

Leavitt path algebragraded ringno-exit graphlocal unitsgraded noetheriangraded artinian

MSC classification

Primary: 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
Secondary: 16W50: Graded rings and modules 16W10: Rings with involution; Lie, Jordan and other nonassociative structures 16D25: Ideals 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 105 , Issue 2 , October 2018 , pp. 229 - 256
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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