Advertisement

Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

  • Daniel LännströmEmail author
Open Access
Article
  • 40 Downloads

Abstract

Let G be a group with neutral element e and let \(S=\bigoplus _{g \in G}S_{g}\) be a G-graded ring. A necessary condition for S to be noetherian is that the principal component Se is noetherian. The following partial converse is well-known: If S is strongly-graded and G is a polycyclic-by-finite group, then Se being noetherian implies that S is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain characterizations of noetherian and artinian Leavitt path algebras with coefficients in a general unital ring. This extends a recent characterization by Steinberg for Leavitt path algebras with coefficients in a commutative unital ring and previous characterizations by Abrams, Aranda Pino and Siles Molina for Leavitt path algebras with coefficients in a field. Secondly, we obtain characterizations of noetherian and artinian unital partial crossed products.

Keywords

Group graded ring Epsilon-strongly graded ring Chain conditions Leavitt path algebra Partial crossed product 

Mathematics Subject Classification (2010)

16P20 16P40 16W50 16S35 

Notes

Acknowledgements

This research was partially supported by the Crafoord Foundation (grant no. 20170843). The author is grateful to Johan Öinert, Stefan Wagner and Patrik Nystedt for giving comments and feedback that helped to improve this manuscript.

References

  1. 1.
    Abrams, G., Aranda Pino, G., Siles Molina, M.: Finite-dimensional Leavitt path algebras. J. Pure Appl. Algebra 209(3), 753–762 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abrams, G., Ara, P., Molina, M.S.: Leavitt Path Algebras, vol. 2191. Springer (2017)Google Scholar
  3. 3.
    Abrams, G., Aranda Pino, G, Siles Molina, M: Locally finite Leavitt path algebras. Israel J. Math. 165(1), 329–348 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Abrams, G., Pino, G.A.: The Leavitt path algebra of a graph. J. Algebra 293 (2), 319–334 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Abrams, G., Pino, G.A., Perera, F., Molina, M.S.: Chain conditions for Leavitt path algebras. In: Forum Mathematicum, vol. 22, pp 95–114. Walter de Gruyter GmbH & Co. KG (2010)Google Scholar
  6. 6.
    Aranda Pino, G, Vaš, L.: Noetherian leavitt path algebras and their regular algebras. Mediterr. J. Math. 10(4), 1633–1656 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bagio, D., Lazzarin, J., Paques, A.: Crossed products by twisted partial actions: Separability, semisimplicity, and frobenius properties. Commun. Algebra 38(2), 496–508 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Batista, E.: Partial actions: What they are and why we care. arXiv:1604.06393 (2016)
  9. 9.
    Bell, A.D: Localization and ideal theory in noetherian strongly group-graded rings. J Algebra 105(1), 76–115 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brown, K.A, Gilmartin, P.: Hopf algebras under finiteness conditions. arXiv:1405.4105 (2014)
  11. 11.
    Carvalho, P.A.A.B., Cortes, W., Ferrero, M.: Partial skew group rings over polycyclic by finite groups. Algebras Represent. Theory 14(3), 449–462 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Clark, L.O., Exel, R., Pardo, E.: A generalized uniqueness theorem and the graded ideal structure of steinberg algebras. In: Forum Mathematicum, vol. 30, pp. 533–552. De Gruyter (2018)Google Scholar
  13. 13.
    Connell, I.G.: On the group ring. Canad. J. Math 15(49), 650–685 (1963)zbMATHCrossRefGoogle Scholar
  14. 14.
    Dokuchaev, M: Partial actions: A survey. Contemp. Math 537, 173–184 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dokuchaev, M, Exel, R, Simón, J.J.: Crossed products by twisted partial actions and graded algebras. J. Algebra 320(8), 3278–3310 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dokuchaev, M, Exel, R.: Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans. Am. Math. Soc. 357(5), 1931–1952 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Exel, R.: Circle actions on c*-algebras, partial automorphisms, and a generalized pimsner-voiculescu exact sequence. J. Funct. Anal. 122(2), 361–401 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Filippov, VT, Kharchenko, VK, Shestakov, IP: The dniester notebook. Unsolved Problems in Ring Theory (1993)Google Scholar
  19. 19.
    Hall, P.: Finiteness conditions for soluble groups. Proc. Lond. Math. Soc. 3(1), 419–436 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hazrat, R.: The graded structure of Leavitt path algebras. Israel J. Math. 195 (2), 833–895 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hazrat, R.: Graded Rings and Graded Grothendieck Groups, vol. 435. Cambridge University Press (2016)Google Scholar
  22. 22.
    Ivanov, S.V.: Group rings of noetherian groups. Math. Notes Acad. Sci. USSR 46(6), 929–933 (1989)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Katsov, Y., Nam, T.G., Zumbrägel, J.: Simpleness of Leavitt path algebras with coefficients in a commutative semiring. In: Semigroup Forum, vol. 94, pp. 481–499. Springer (2017)Google Scholar
  24. 24.
    Lam, T.-Y.: Lectures on Modules and Rings, vol. 189. Springer Science & Business Media (2012)Google Scholar
  25. 25.
    Larki, H.: Ideal structure of Leavitt path algebras with coefficients in a unital commutative ring. Commun. Algebra 43(12), 5031–5058 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lännström, D.: Induced quotient gradings of epsilon-strongly graded rings. In preperation (2018)Google Scholar
  27. 27.
    Nastasescu, C., van Oystaeyen, F.: Methods of Graded Rings Lecture Notes in Mathematics. Springer (2004)Google Scholar
  28. 28.
    Nystedt, P., Öinert, J., Pinedo, H.: Artinian and noetherian partial skew groupoid rings. J. Algebra 503, 433–452 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Nystedt, P., Öinert, J., Pinedo, H.: Epsilon-strongly graded rings, separability and semisimplicity. J. Algebra 514, 1–24 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Nystedt, P., Öinert, J.: Epsilon-strongly graded Leavitt path algebras. arXiv:1703.10601 (2017)
  31. 31.
    Park, J.K.: Artinian skew group rings. Proc. Am. Math. Soc. 75(1), 1–7 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Passman, D.S.: Radicals of twisted group rings. Proc. Lond. Math. Soc. 3(3), 409–437 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Saorín, M: Descending chain conditions for graded rings. Proc. Am. Math. Soc. 115(2), 295–301 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Steinberg, B.: Chain conditions on étale groupoid algebras with applications to Leavitt path algebras and inverse semigroup algebras. J. Aust. Math. Soc., 1–9 (2018)Google Scholar
  35. 35.
    Tomforde, M.: Leavitt path algebras with coefficients in a commutative ring. arXiv:0905.0478 (2009)

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics and Natural SciencesBlekinge Institute of TechnologyKarlskronaSweden

Personalised recommendations