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STABLE RANK OF LEAVITT PATH ALGEBRAS OF ARBITRARY GRAPHS

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  12 December 2012

HOSSEIN LARKI  and
ABDOLHAMID RIAZI
HOSSEIN LARKI*
Affiliation:
Department of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University, Ahwaz, Iran
ABDOLHAMID RIAZI
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology,424 Hafez Ave., 15914 Tehran, Iran email [email protected]
*
✉[email protected][email protected]
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Abstract

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The stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this to an arbitrary directed graph. In part our computation proceeds as for the row-finite case, but we also use knowledge of the row-finite setting by applying the desingularising method due to Drinen and Tomforde. In particular, we characterise purely infinite simple quotients of a Leavitt path algebra.

Keywords

Leavitt path algebrastable rankpurely infinite quotient

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 206 - 217
Copyright
Copyright ©2012 Australian Mathematical Publishing Association Inc. 

References

Abrams, G. and Aranda Pino, G., ‘The Leavitt path algebras of a graph’, J. Algebra 293 (2005), 319–334.CrossRefGoogle Scholar
Abrams, G. and Aranda Pino, G., ‘Purely infinite simple Leavitt path algebras’, J. Pure Appl. Algebra 207 (2006), 553–563.CrossRefGoogle Scholar
Abrams, G. and Aranda Pino, G., ‘The Leavitt path algebras of arbitrary graphs’, Houston J. Math. 34 (2008), 323–442.Google Scholar
Abrams, G., Aranda Pino, G., Perera, F. and Siles Molina, M., ‘Chain conditions for Leavitt algebras’, Forum Math. 22 (1) (2010), 95–114.CrossRefGoogle Scholar
Abrams, G., Aranda Pino, G. and Siles Molina, M., ‘Finite dimensional Leavitt path algebras’, J. Pure Appl. Algebra 209 (2007), 753–762.CrossRefGoogle Scholar
Abrams, G., Aranda Pino, G. and Siles Molina, M., ‘Locally finite Leavitt path algebras’, Israel J. Math. 165 (2008), 329–348.CrossRefGoogle Scholar
Ara, P., Goodearl, K. R. and Pardo, E., ‘${K}_{0} $ of purely infinite simple regular rings’, K-Theory 26 (7) (2002), 69–100.CrossRefGoogle Scholar
Ara, P., Moreno, M. A. and Pardo, E., ‘Nonstable K-theory for graph algebras’, Algebr. Represent. Theory 10 (2007), 157–178.CrossRefGoogle Scholar
Ara, P. and Pardo, E., ‘Stable rank of Leavitt path algebras’, Proc. Amer. Math. Soc. 136 (7) (2008), 2375–2386.CrossRefGoogle 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. Iberoam. 26 (2) (2010), 611–638.CrossRefGoogle Scholar
Aranda Pino, G., Pardo, E. and Siles Molina, M., ‘Exchanging Leavitt path algebras and stable rank’, J. Algebra 305 (2006), 912–936.CrossRefGoogle Scholar
Bates, T., Hong, J. H., Raeburn, I. and Symański, W., ‘The ideal structure of the ${C}^{\ast } $-algebras of infinite graphs’, Ilinois J. Math. 46 (2002), 1159–1176.Google Scholar
Bates, T., Pask, D., Raeburn, I. and Symański, W., ‘The ${C}^{\ast } $-algebras of row-finite graphs’, New York J. Math. 6 (2000), 307–324.Google Scholar
Deicke, K., Hong, J. H. and Szymański, W., ‘Stable rank of graph algebras. Type I graph algebras and their limits’, Indiana Univ. Math. J. 52 (2003), 963–979.CrossRefGoogle Scholar
Drinen, D. and Tomforde, M., ‘The ${C}^{\ast } $-algebras of arbitrary graphs’, Rocky Mountain J. Math. 35 (2005), 105–135.CrossRefGoogle Scholar
Leavitt, W. G., ‘The module type of a ring’, Trans. Amer. Math. Soc. 103 (1962), 113–130.CrossRefGoogle Scholar
McConnell, J. C. and Robson, J. C., Noncommutative Noetherian Rings (John Wiley & Sons, New York, 1982).Google Scholar
Raeburn, I., Graph Algebras (American Mathematical Society, Providence, RI, 2005).CrossRefGoogle Scholar
Ruiz, E. and Tomforde, M., ‘Ideals in graph algebras’, Preprint, arXiv:1205.1247.Google Scholar
Tomforde, M., ‘Stability of ${C}^{\ast } $-algebras associated to graphs’, Proc. Amer. Math. Soc. 132 (6) (2004), 1787–1795.CrossRefGoogle Scholar
Tomforde, M., ‘Uniqueness theorems and ideal structure for Leavitt path algebras’, J. Algebra 318 (2007), 270–299.CrossRefGoogle Scholar
Vaserstein, L. N., ‘Stable rank of rings and dimensionality of topological spaces’, Funct. Anal. Appl. 5 (1971), 102–110.CrossRefGoogle Scholar