Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-20T01:38:29.412Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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. 

References

Abrams, G., Aranda Pino, G., Perera, F. and Siles Molina, M., ‘Chain conditions for Leavitt path algebras’, Forum Math. 22 (2010), 95–114.Google Scholar
Abrams, G. and Rangaswamy, K. M., ‘Regularity conditions for arbitrary Leavitt path algebras’, Algebr. Represent. Theory 13 (2010), 319–334.Google Scholar
Abrams, G. and Tomforde, M., ‘Isomorphism and Morita equivalence of graph algebras’, Trans. Amer. Math. Soc. 363(7) (2011), 3733–3767.Google Scholar
Ara, P. and Goodearl, K. R., ‘Leavitt path algebras of separated graphs’, J. reine angew. Math. 669 (2012), 165–224.Google Scholar
Ara, P., Moreno, M. A. and Pardo, E., ‘Nonstable K-theory for graph algebras’, Algebr. Represent. Theory 10(2) (2007), 157–178.Google Scholar
Aranda Pino, G., Brox, J. and Siles Molina, M., ‘Cycles in Leavitt path algebras by means of idempotents’, Forum Math. 27(1) (2015), 601–633.Google Scholar
Aranda Pino, G., Martín Barquero, D., Martín González, C. and Siles Molina, M., ‘Socle theory for Leavitt path algebras of arbitrary graphs’, Rev. Mat. Iberoamericana 26(2) (2010), 611–638.Google Scholar
García, J. L. and Simón, J. J., ‘Morita equivalence for idempotent rings’, J. Pure Appl. Algebra 76 (1991), 39–56.Google Scholar
Goodearl, K. R., Von Neumann Regular Rings, 2nd edn (Krieger, Malabar, FL, 1991).Google Scholar
Goto, S. and Yamagishi, K., ‘Finite generation of noetherian graded rings’, Proc. Amer. Math. Soc. 89(1) (1983), 41–44.Google Scholar
Hazrat, R., ‘The graded structure of Leavitt path algebras’, Israel J. Math. 195 (2013), 833–895.Google Scholar
Hazrat, R., ‘Leavitt path algebras are graded von Neumann regular rings’, J. Algebra 401 (2014), 220–233.Google Scholar
Hazrat, R., Graded Rings and Graded Grothendieck Groups, London Mathematical Society Lecture Note Series, 435 (Caimbridge University Press, Cambridge, 2016).Google Scholar
Hazrat, R. and Rangaswamy, K., ‘On graded irreducible representations of Leavitt path algebras’, J. Algebra 450 (2016), 458–486.Google Scholar
Hazrat, R. and Vaš, L., ‘ $K$ -theory classification of graded ultramatricial algebras with involution’, Preprint, 2016, arXiv:1604.07797.Google Scholar
Hazrat, R. and Vaš, L., ‘Baer and Baer ∗-ring characterizations of Leavitt path algebras’, J. Pure Appl. Algebra 222(1) (2018), 39–60.Google Scholar
Lozano, M. G. and Molina, M. S., ‘Quotient rings and Fountain–Gould left orders by the local approach’, Acta Math. Hungar. 97 (2002), 287–301.Google Scholar
Mesyan, Z. and Vaš, L., ‘Traces on semigroup rings and Leavitt path algebras’, Glasg. Math. J. 58(1) (2016), 97–118.Google Scholar
Muhly, P. S. and Tomforde, M., ‘Adding tails to C ∗ -correspondences’, Doc. Math. 9 (2004), 79–106.Google Scholar
Năstăsescu, C. and van Oystaeyen, F., Methods of Graded Rings, Lecture Notes in Mathematics, 1836 (Springer, Berlin, 2004).Google Scholar
Tomforde, M., ‘Uniqueness theorems and ideal structure for Leavitt path algebras’, J. Algebra 318(1) (2007), 270–299.Google Scholar
Vaš, L., ‘Canonical traces and directly finite Leavitt path algebras’, Algebr. Represent. Theory 18 (2015), 711–738.Google Scholar