Skip to main content
SpringerLink
Log in

Burnside-type problems, theorems on height, and independence

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

This review paper is devoted to some questions related to investigations of bases in PI-algebras. The central point is generalization and refinement of the Shirshov height theorem, of the Amitsur–Shestakov hypothesis, and of the independence theorem. The paper is mainly inspired by the fact that these topics shed some light on the analogy between structure theory and constructive combinatorial reasoning related to the “microlevel,” to relations in algebras and straightforward calculations. Together with the representation theory of monomial algebras, height and independence theorems are closely connected with combinatorics of words and of normal forms, as well as with properties of primary algebras and with combinatorics of matrix units. Another aim of this paper is an attempt to create a kind of symbolic calculus of operators defined on records of transformations.

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. S. A. Amitsur, “A generalization of Hilbert’s Nullstellensatz,” Proc. Amer. Math. Soc., 8, No. 4, 649–656 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Amitsur, “Rational identities and applications to algebra and geometry,” J. Algebra, 3, No. 3, 304–359 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Amitsur, “The sequence of codimensions of PI-algebras,” Israel J. Math., 47, No. 1, 1–22 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  4. K. I. Beidar, V. N. Latyshev, V. T. Markov, A. V. Mikhalev, L. A. Skornyakov, and A. A. Tuganbaev, “Associative rings,” J. Sov. Math., 38, 1855–1929 (1987).

    Article  MATH  Google Scholar 

  5. A. Ya. Belov, “On a Shirshov base with respect to free algebras of complexity n,” Mat. Sb., 135 (177), No. 3, 373–384 (1988).

    MathSciNet  Google Scholar 

  6. A. Belov, “Some estimations for nilpotence of nil-algebras over a field of an arbitrary characteristics and height theorem,” Comm. Algebra, 20, No. 10, 2919–2922 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Belov, “About height theorem. Comm. Algebra, 23, No. 9, 3551–3553 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  8. A.Ya. Belov, “The Nagata–Higman theorem for semirings,” Fundam. Prikl. Mat., 1, No. 2, 523–527 (1995).

    MATH  MathSciNet  Google Scholar 

  9. A. Belov, V. Borisenko, and V. Latyshev, “Monomial algebras,“ J. Math. Sci., 87, No. 3, 3463–3575 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Berele, “Homogeneous polynomial identities,” Israel J. Math., 42, No. 3, 258–272 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  11. L. A. Bokut’ and I. P. Shestakov, “Some results by A. I. Shirshov and his school,” in: Bokut’ L. A. (ed.) et al., Proc. Second Int. Conf. on Algebra Dedicated to the Memory of A. I. Shirshov, August 20–25, 1991, Barnaul, Russia,, Contemp. Math., Vol. 184, Amer. Math. Soc., Providence (1995), pp. 1–12.

    Google Scholar 

  12. A. Braun, “The radical in a finitely generated PI-algebra,” Bull. Amer. Math. Soc., 7, No. 2, 385–386 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  13. G. P. Chekanu, Locally Finite Algebras, Ph.D. Thesis, Kishinev (1982).

  14. G. P. Chekanu, “On Shirshov’s height theorem,” in: Proc. of XIX All-Union Alg. Conf. I, L’vov (1987), p. 306.

  15. G. P. Chekanu, “On local finiteness of algebras,” Mat. Issled., 105, 153–171 (1988).

    MATH  MathSciNet  Google Scholar 

  16. G. P. Chekanu, “Independence and quasiregularity in algebras,” in: Third Int. Conf. on Algebra Dedicated to the Memory of M. I. Kargapolov, Krasnoyarsk (1993).

  17. G. P. Chekanu, “Independence and quasiregularity in algebras,” Dokl. Akad. Nauk, 337, No. 3, 316–319 (1994).

    Google Scholar 

  18. G. P. Chekanu, Independence and Local Finiteness, Doctor Habilitat Thesis, Kishinev (1995).

  19. Dnestr Notebook, Inst. Mat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1993).

  20. V. Drensky, “Codimensions of T-ideals and Hilbert series of relatively free algebras,” J. Algebra, 91, No. 1, 1–17 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Drensky, “On the Hilbert series of relatively free algebras,” Comm. Algebra, 12, No. 19, 2335–2347 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  22. V. S. Drensky, Free Algebras and PI-Algebras, Springer Singapore, Singapore (2000).

    MATH  Google Scholar 

  23. E. Formanek, P. W. Halpin, and C. W. Li, “The Poincare series of the ring of 2 × 2 generic matrices,” J. Algebra, 69, 105–112 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Giambruno and M. Zaicev, “Exponential codimension growth of PI algebras: An exact estimate,” Adv. Math., 142, 221–243 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Giambruno and M. Zaicev, “Minimal varieties of algebras of exponential growth,” Electron. Res. Announc. Amer. Math. Soc., 6, 40–44 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  26. A. V. Grishin, “The growth exponent of a variety of algebras and its applications,” Algebra Logika, 28, No. 5, 536–557 (1987).

    MathSciNet  Google Scholar 

  27. I. N. Herstein, Noncommutative rings, Carus Math. Monographs, No. 15, Math. Assoc. Amer. (1968).

  28. A. R. Kemer, Nonmatrix Varieties, Varieties with Polynomial Growth, and Finitely Generated PI-Algebras, Ph.D. Thesis, Novosibirsk (1981).

  29. A. R. Kemer, “Varieties and \( \mathbb{Z}\)2-graded algebras,” Izv. Akad. Nauk SSSR, Ser. Mat., 48, No. 5, 1042–1059 (1984).

    MathSciNet  Google Scholar 

  30. A. T. Kolotov, “An upper estimate on the height of a finitely generated algebra with identical relations,” Sib. Mat. Zh., 23, No. 1, 187–189 (1982).

    MATH  MathSciNet  Google Scholar 

  31. A. I. Kostrikin, Around Burnside, Springer (1990).

  32. The Kourovka Notebook (Unsolved Problems in Group Theory), Inst. Mat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1995).

  33. A. G. Kurosh, “Problems of ring theory connected with the Burnside problem on periodic groups,” Izv. Akad. Nauk SSSR. Ser. Mat., 5, 233–240 (1941).

    Google Scholar 

  34. I. V. L’vov, “On Shirshov’s height theorem,” in: Fifth All-Union Sympos. Theory of Rings, Algebras, and Modules, Abstracts of Reports[in Russian], Inst. Mat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1982).

    Google Scholar 

  35. Yu. A. Medvedev, Nilradicals of Finitely Generated Jordan PI-Algebras [in Russian], Preprint No. 24, Inst. Mat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1985).

    Google Scholar 

  36. S. P. Mishchenko, “A variant of the height theorem for Lie algebras,” Math. Notes, 47, No. 4, 368–372 (1990).

    MATH  MathSciNet  Google Scholar 

  37. S. V. Pchelintsev, “The height theorem for alternate algebras,” Mat. Sb., 124, No. 4, 557–567 (1984).

    MathSciNet  Google Scholar 

  38. C. Procesi, Rings with Polynomial Identities, Marcel Dekker, New York (1973).

    MATH  Google Scholar 

  39. Yu. P. Razmyslov, “Algebras satisfying identity relations of Capelli type,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, No. 1, 143–166 (1981).

    MATH  MathSciNet  Google Scholar 

  40. Yu. P. Razmyslov, Identities of Algebras and Their Representations [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  41. L. H. Rowen, Polynomial Identities in Ring Theory, Pure Appl. Math., Vol. 84, Academic Press, New York (1980).

    Google Scholar 

  42. I. P. Shestakov, “Finitely generated special Jordan algebras and alternative PI-algebras,” Mat. Sb., 122, No. 1, 31–40 (1983).

    MathSciNet  Google Scholar 

  43. A. I. Shirshov, “On some nonassociative nil-rings and algebraic algebras,” Mat. Sb., 41, No. 3, 381–394 (1957).

    MathSciNet  Google Scholar 

  44. A. I. Shirshov, “On rings with identity relations,” Mat. Sb., 43, No. 2, 277–283 (1957).

    MathSciNet  Google Scholar 

  45. V. G. Skosyrskii, Jordan Algebras with the Minimality Condition for Two-Sided Ideals [in Russian], Preprint No. 21, Inst. Mat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1985).

    Google Scholar 

  46. L. W. Small, “Localization in PI-rings,” J. Algebra, 19, No. 2, 181–183 (1971).

    MathSciNet  Google Scholar 

  47. W. Specht, “Gesetze in Ringen,” Math. Z., 52, No. 5, 557–589 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  48. V. A. Ufnarovskij, “An independence theorem and its consequences,” Mat. Sb., 128 (170), No. 1 (9), 124–132 (1985).

    Google Scholar 

  49. V. A. Ufnarovskij, “Combinatorial and asymptotic methods in algebra,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions, [in Russian], Vol. 57, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1990), pp. 5–177.

    Google Scholar 

  50. E. I. Zel’manov, “Absolute zero-divisors and algebraic Jordan algebras,” Sib. Mat. Zh., 23, No. 6, 100–116 (1982).

    MathSciNet  Google Scholar 

  51. K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative, Academic Press (1982).

  52. K. A. Zubrilin, “Algebras satisfying Capelli identities,” Mat. Sb., 186, No. 3, 53–64 (1995).

    MathSciNet  Google Scholar 

  53. K. A. Zubrilin, “On the class of nilpotency of obstruction to embeddability of algebras satisfying Capelli identities,” Fundam. Prikl. Mat., 1, No. 2, 409–430 (1995).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Open Education, Moscow, Russia

    A. Ya. Belov

  2. International University of Bremen, Bremen, Russia

    A. Ya. Belov

  3. Hebrew University of Jerusalem, Jerusalem, Israel

    A. Ya. Belov

Authors
  1. A. Ya. Belov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ya. Belov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 19–79, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belov, A.Y. Burnside-type problems, theorems on height, and independence. J Math Sci 156, 219–260 (2009). https://doi.org/10.1007/s10958-008-9264-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9264-3

Keywords

Search

Navigation