Skip to main content
SpringerLink
Log in

The inversion height of the free field is infinite

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

Let \(X\) be a finite set with at least two elements, and let \(k\) be any commutative field. We prove that the inversion height of the embedding \(k\langle X\rangle \hookrightarrow D\), where \(D\) denotes the universal (skew) field of fractions of the free algebra \(k\langle X\rangle \), is infinite. Therefore, if \(H\) denotes the free group on \(X\), the inversion height of the embedding of the group algebra \(k H\) into the Malcev–Neumann series ring is also infinite. This answers in the affirmative a question posed by Neumann (Trans Am Math Soc 66:202–252, 1949). We also give an infinite family of examples of non-isomorphic fields of fractions of \(k\langle X\rangle \) with infinite inversion height. We show that the universal field of fractions of a crossed product of a field by the universal enveloping algebra of a free Lie algebra is a field of fractions constructed by Cohn (and later by Lichtman). This extends a result by A. Lichtman.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amayo, R.K., Stewart, I.: Infinite-Dimensional Lie Algebras. Noordhoff International Publishing, Leyden (1974)

    Book  MATH  Google Scholar 

  2. Baumslag, B.: Free Lie algebras and free groups. J. Lond. Math. Soc. 2(4), 523–532 (1972)

    Article  MathSciNet  Google Scholar 

  3. Bergen, J., Passman, D.S.: Delta methods in enveloping rings. J. Algebra 133, 277–312 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bergman, G.M.: Modules over coproducts of rings. Trans. Am. Math. Soc. 200, 1–32 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications, Encyclopedia of Mathematics and its Applications, vol. 137. Cambridge University Press, Cambridge (2011)

  6. Bokut’, L.A., Kukin, G.P.: Algorithmic and Combinatorial Algebra, Mathematics and Its Applications, vol. 255. Kluwer Academic Publishers, Dordrecht (1993)

    Google Scholar 

  7. Chin, W.: Prime ideals in differential operator rings and crossed products of infinite groups. J. Algebra 106(1), 78–104 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cho, E.-H., Oh, S.-Q.: Skew enveloping algebras and Poisson enveloping algebras. Commun. Korean Math. Soc. 20(4), 649–655 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cohn, P.M.: Rings with a weak algorithm. Trans. Am. Math. Soc. 109, 332–356 (1963)

    Article  MATH  Google Scholar 

  10. Cohn, P.M.: An embedding theorem for free associative algebras. Math. Pannon. 1(1), 49–56 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Cohn, P.M.: Skew Fields. Theory of General Division Rings, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge (1995)

  12. Cohn, P.M.: Free Ideal Rings and Localization in General Rings, New Mathematical Monographs, vol. 3. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  13. Dicks, W., Dunwoody, M.J.: Groups Acting on Graphs, Cambridge Studies in Advanced Mathematics, vol. 17. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  14. Gelfand, I., Gelfand, S., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193(1), 56–141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goodearl, K.R., Warfield Jr, R.B.: An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, vol. 16. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  16. Herbera, D., Sánchez, J.: Computing the inversion height of some embeddings of the free algebra and the free group algebra. J. Algebra 310(1), 108–131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Higman, G.: The units of group-rings. Proc. Lond. Math. Soc. (2) 46, 231–248 (1940)

    Article  MathSciNet  Google Scholar 

  18. Hughes, I.: Division rings of fractions for group rings. Commun. Pure Appl. Math. 23, 181–188 (1970)

    Article  MATH  Google Scholar 

  19. Hughes, I.: Division rings of fractions for group rings. II. Commun. Pure Appl. Math. 25, 127–131 (1972)

    Article  MATH  Google Scholar 

  20. Kurosh, A.G.: The theory of groups. Chelsea Publishing Co., New York (1960), Translated from the Russian and edited by K. A. Hirsch. 2nd English ed. 2 volumes

  21. Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131, 2nd edn. Springer, New York (2001)

    Book  Google Scholar 

  22. Lewin, J.: Fields of fractions for group algebras of free groups. Trans. Am. Math. Soc. 192, 339–346 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lewin, J., Lewin, T.: An embedding of the group algebra of a torsion-free one-relator group in a field. J. Algebra 52(1), 39–74 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lichtman, A.I.: On matrix rings and linear groups over fields of fractions of group rings and enveloping algebras. II. J. Algebra 90(2), 516–527 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lichtman, A.I.: Valuation methods in division rings. J. Algebra 177(3), 870–898 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lichtman, A.I.: On universal fields of fractions for free algebras. J. Algebra 231(2), 652–676 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Makar-Limanov, L., Malcolmson, P.: Free subalgebras of enveloping fields. Proc. Am. Math. Soc. 111(2), 315–322 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Makar-Limanov, L., Umirbaev, U.: Free Poisson Fields and Their Automorphisms, arXiv:1202.4382v1

  29. Mal’cev, A.I.: On the embedding of group algebras in division algebras. Dokl. Akad. Nauk SSSR (N.S.) 60, 1499–1501 (1948)

    MathSciNet  Google Scholar 

  30. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings, Pure and Applied Mathematics (New York). Wiley, Chichester. With the cooperation of L.W. Small, A Wiley-Interscience Publication (1987)

  31. Montgomery, Susan.: Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC (1993)

  32. Neumann, B.H.: On ordered division rings. Trans. Am. Math. Soc. 66, 202–252 (1949)

    Article  MATH  Google Scholar 

  33. Donald, S.: Passman, Infinite Crossed Products, Pure and Applied Mathematics, vol. 135. Academic Press Inc., Boston, MA (1989)

    Google Scholar 

  34. Reutenauer, C.: Inversion height in free fields. Selecta Math. (N.S.) 2(1), 93–109 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Reutenauer, C.: Malcev–Neumann series and the free field. Exposition. Math. 17(5), 469–478 (1999)

    MathSciNet  MATH  Google Scholar 

  36. Sánchez, J.: Localization: On Division Rings and Tilting Modules, D.Phil. Thesis, Universitat Autònoma de Barcelona (2008)

  37. Sánchez, J.: Free group algebras in Malcev–Neumann skew fields of fractions. Forum Math. 26(2), 443–466 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Shalev, A., Zelmanov, E.I.: Narrow Lie algebras: a coclass theory and a characterization of the Witt algebra. J. Algebra 189(188), 294–331 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Both authors are grateful to Bill Chin for pointing out suitable references on Lie algebra crossed products. They also thank Yago Antolín for providing them with examples that show the limits of the construction of Proposition 5.3. Last but not least, the authors are grateful to the referee for the extremely careful reading of the paper, the correction of a gap, and many interesting suggestions and comments. The second author would like to thank Alexander Lichtman for interesting conversations on the papers [25, 26] and related topics. He also thanks the Mathematics Department at the University of Wisconsin Parkside, where these conversations took place, for its kind hospitality during his visit.

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    Dolors Herbera

  2. Department of Mathematics - IME, University of São Paulo, Caixa Postal 66281, São Paulo, SP, 05314-970, Brazil

    Javier Sánchez

Authors
  1. Dolors Herbera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Javier Sánchez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Javier Sánchez.

Additional information

The second named author was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) processo número 2009/50886-0.

Both authors acknowledge partial support from DGI MINECO MTM2011-28992-C02-01, by ERDF UNAB10-4E-378 “A way to build Europe”, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya Project 2009 SGR 1389.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Herbera, D., Sánchez, J. The inversion height of the free field is infinite. Sel. Math. New Ser. 21, 883–929 (2015). https://doi.org/10.1007/s00029-014-0168-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-014-0168-4

Mathematics Subject Classification

Search

Navigation