Skip to main content
SpringerLink
Log in

The radical RN and the weakly solvable radical of linear groups over associative rings

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

Abstract

This paper is devoted to the computation of the radicals RN and RN* and the weakly solvable radical for a number of basic classical linear groups over rings, including the unitary group over a ring with involution and matrix groups normalized by elementary Chevalley groups.

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. E. Abe and J. Hurley, “Centers of Chevalley groups over commutative rings,” Commun. Algebra, 16, No. 1, 57–74 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  2. V. A. Andrunakievich and V. M. Ryabukhin, Radicals of Algebras and the Structure Theory [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  3. R. Baer, “Nilgruppen,” Math. Z., 62, No. 4, 402–437 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  4. K. I. Beidar and S. A. Pikhtilkov, “On the prime radical of the special Lie algebras,” Usp. Mat. Nauk, 49, No. 1, 233 (1994).

    MathSciNet  Google Scholar 

  5. K. I. Beidar and S. A. Pikhtilkov, “The prime radical of the special Lie algebras,” Fundam. Prikl. Mat., 6, No. 3, 643–648 (2000).

    MATH  MathSciNet  Google Scholar 

  6. L. A. Bokut', Associative Rings [in Russian], Vols. 1, 2, Izd. Novosibirsk. Univ., Novosibirsk (1977, 1981).

    Google Scholar 

  7. A. Borel, Linear Algebraic Groups, W. A. Benjamin, New York (1969).

    MATH  Google Scholar 

  8. A. Borel, “Properties and linear representations of Chevalley groups,” in: Semin. Algebr. Groups related Finite Groups. Princeton 1968/69, Lect. Notes Math., Vol. 131, Springer, Berlin (1970), pp. 1–55.

    Chapter  Google Scholar 

  9. A. Braun, “The nilpotency of the radical in a finitely generated PI-ring,” J. Algebra, 89, No. 2, 375–396 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  10. V. H. Chan, “On the minimal radical class over class of Abelian groups,” Dokl. Akad. Nauk SSSR, 149, No. 6, 1270–1273 (1963).

    Google Scholar 

  11. V. H. Chan, “Nil-groups of a finite rank,” Sib. Mat. Zh., 5, No. 2, 459–464 (1964).

    Google Scholar 

  12. C. Chevalley, Certains schemas de groupes semi-simples, Sémin. Bourbaki (1960/61), No. 219, Secretariat Mathematique, Paris (1961).

    Google Scholar 

  13. M. Demazure, “Schemas en groupes reductifs,” Bull. Soc. Math. France, 93, 369–413 (1965).

    MATH  MathSciNet  Google Scholar 

  14. J. D. Dixon, “The solvable length of solvable linear group,” Math. Z., 107, No. 2, 151–158 (1968).

    Article  MathSciNet  Google Scholar 

  15. I. Z. Golubchik and A. V. Mikhalev, “Isomorphisms of unitary groups over associative rings,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov, 132, 97–109 (1983).

    MathSciNet  Google Scholar 

  16. I. Z. Golubchik and A. V. Mikhalev, “Elementary subgroup of unitary group over PI-ring,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 30–36 (1985).

  17. A. Yu. Golubkov, “Prime radical of Chevalley group over commutative ring,” Usp. Mat. Nauk, 55, No. 3, 173–174 (2000).

    MathSciNet  Google Scholar 

  18. A. Yu. Golubkov, “The prime radical of Chevalley group over commutative ring,” J. Math. Sci., 102, No. 6, 4781–4790 (2000).

    MathSciNet  Google Scholar 

  19. A. Yu. Golubkov, “Prime (RI*-soluble) radical of unitary group over ring with involution,” Fundam. Prikl. Mat., 6, No. 1, 93–119 (2000).

    MATH  MathSciNet  Google Scholar 

  20. A. Yu. Golubkov, “Prime radical of adjoint Chevalley group over associative ring,” Usp. Mat. Nauk, 56, No. 4, 143–144 (2001).

    MathSciNet  Google Scholar 

  21. A. Yu. Golubkov, Prime Radical of Classical Groups over Associative Rings [in Russian], Candidate's Dissertation in Physics and Mathematics, Moscow (2001).

  22. A. Yu. Golubkov, “The prime radical of the special Lie algebras and the elementary Chevalley groups,” Comm. Algebra, 32, No. 5, 1649–1683 (2004).

    MATH  MathSciNet  Google Scholar 

  23. A. Yu. Golubkov, A. V. Mikhalev, and S. A. Pikhtilkov, “Classical nil-radicals of Lie algebras” (in preparation).

  24. I. N. Herstein, Noncommutative Rings, Carus Math. Mon., Math. Assoc. Amer., Washington (1996).

    MATH  Google Scholar 

  25. K. A. Hirsch, “Über lokal-nilpotente Gruppen,” Math. Z., 63, No. 3, 290–294 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  26. J. E. Humphreys, Linear Algebraic Groups, Springer, Berlin (1975).

    MATH  Google Scholar 

  27. N. Jacobson, Structure of Rings, Colloquium Publications, Vol. 37, Amer. Math. Soc., Providence (1964).

    Google Scholar 

  28. N. Jacobson, Lie Algebras, Interscience, New York (1962).

    MATH  Google Scholar 

  29. A. R. Kemer, “Capelli identities and nilpotence of a radical of finitely generated PI-algebra,” Dokl. Akad. Nauk SSSR, 255, No. 4, 793–797 (1980).

    MathSciNet  Google Scholar 

  30. Sh. S. Kemkhadze, “On stable groups of automorphisms,” Dokl. Akad. Nauk SSSR, 158, No. 3, 510–512 (1964).

    MathSciNet  Google Scholar 

  31. Sh. S. Kemkhadze, “Quasi-nilpotent groups,” Dokl. Akad. Nauk SSSR, 155, No. 5, 1003–1005 (1964).

    MathSciNet  Google Scholar 

  32. I. S. Klein and A. V. Mikhalev, “Orthogonal Steinberg group over ring with involution,” Algebra Logika, 9, No. 2, 145–166 (1970).

    Google Scholar 

  33. A. I. Kostrikin, Around Burnside [in Russian], Nauka, Moscow (1986).

    MATH  Google Scholar 

  34. A. G. Kurosh, Theory of Groups [in Russian], Lan', St. Petersburg (2005).

    Google Scholar 

  35. A. G. Kurosh, “Radicals in group theory,” Sib. Mat. Zh., 3, No. 6, 912–931 (1962).

    Google Scholar 

  36. J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham (1966).

    MATH  Google Scholar 

  37. I. V. Lvov, Braun's Theorem of a Radical of Finitely Generated PI-Algebra [in Russian], Preprint No. 63, Novosibirsk (1984).

  38. A. I. Malcev, “On the isomorphic representation of infinite groups by matrices,” Mat. Sb., 8, No. 3, 405–422 (1940).

    MathSciNet  Google Scholar 

  39. A. I. Malcev, “On the representation of infinite dimensional algebras,” Mat. Sb., 13, Nos. 2–3, 263–285 (1943).

    MathSciNet  Google Scholar 

  40. V. A. Parfenov, “On the weakly solvable radical of Lie algebras,” Sib. Mat. Zh., 12, No. 1, 171–176 (1971).

    Article  MathSciNet  Google Scholar 

  41. D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, New York (1977).

    MATH  Google Scholar 

  42. S. A. Pikhtilkov, “On the prime radical of PI-representable groups,” Mat. Zametki, 72, No. 5, 739–744 (2002).

    MathSciNet  Google Scholar 

  43. S. A. Pikhtilkov, The Structural Theory of Special Lie Algebras [in Russian], Tul'skij Gos. Ped. Univ., Tula (2005).

    Google Scholar 

  44. V. P. Platonov, “Engel elements and the radical in PI-algebras and topological groups,” Dokl. Akad. Nauk SSSR, 161, No. 2, 288–291 (1965).

    MathSciNet  Google Scholar 

  45. B. I. Plotkin, “On some attributes of locally nilpotent groups,” Usp. Mat. Nauk, 9, No. 3 (61), 181–187 (1954).

    MATH  MathSciNet  Google Scholar 

  46. B. I. Plotkin, “Generalized solvable and generalized nilpotent groups,” Usp. Mat. Nauk, 13, No. 4, 90–172 (1958).

    MathSciNet  Google Scholar 

  47. B. I. Plotkin, “On the algebraic sets of elements in groups and Lie algebras,” Usp. Mat. Nauk, 13, No. 6 (84), 133–138 (1958).

    MATH  MathSciNet  Google Scholar 

  48. B. I. Plotkin, “Notes on Engel groups and Engel elements on groups. Some generalizations,” Izv. Uralsk. Gos. Univ., No. 36, 153–166 (2005).

  49. Yu. P. Razmyslov, “On the Jacobson radical in PI-algebras,” Algebra Logika, 13, No. 3, 337–360 (1974).

    MathSciNet  Google Scholar 

  50. L. H. Rowen, “Some results on the center of a ring with polynomial identity,” Bull. Amer. Math. Soc., 79, No. 1, 219–223 (1973).

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  52. K. K. Shchukin, “RI*-solvable radical of groups,” Mat. Sb., 52, No. 4, 1024–1031 (1960).

    Google Scholar 

  53. K. K. Shchukin, “To the radical theory in groups,” Sib. Mat. Zh., 3, No. 6, 932–942 (1962).

    MATH  Google Scholar 

  54. K. K. Shchukin, “To the radical theory in groups,” Dokl. Akad. Nauk SSSR, 142, No. 5, 1047–1049 (1962).

    MathSciNet  Google Scholar 

  55. L. A. Skornyakov, Elements of General Algebra [in Russian], Nauka, Moscow (1983).

    MATH  Google Scholar 

  56. M. R. Stein, “Relativizing functors on rings and algebraic K-theory,” J. Algebra, 19, No. 1, 140–152 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  57. R. Steinberg, Lectures on Chevalley Groups, Mimeographed Lecture Notes, Yale Univ. Math. Dept., New Haven (1967).

    Google Scholar 

  58. D. A. Suprunenko, Matrix Groups [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  59. G. Taddei, “Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau,” in: Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory. Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder, 1983, Pt. II, Contemp. Math., Vol. 55, Amer. Math. Soc., Providence (1986), pp. 693–710.

    Google Scholar 

  60. L. N. Vaserstein, “Stabilization of unitary and orthogonal groups over ring with involution,” Mat. Sb., 81, No. 3, 328–351 (1970).

    MathSciNet  Google Scholar 

  61. L. Vaserstein, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 38, No. 2, 219–230 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  62. N. A. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Proc. Conf. Nonassociative Algebras and Related Topics (Hiroshima-1990), World Scientific, London (1991), pp. 219–335.

    Google Scholar 

  63. H. Zassenhaus, “Beweis eines Satzes über diskrete Gruppen,” Abh. Math. Semin. Hansische Univ., 12, No. 3/4, 289–312 (1938).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics and Control Systems, Bauman Moscow State Technical University, Moscow, Russia

    A. Yu. Golubkov

Authors
  1. A. Yu. Golubkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Yu. Golubkov.

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 31–115, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Golubkov, A.Y. The radical RN and the weakly solvable radical of linear groups over associative rings. J Math Sci 154, 143–203 (2008). https://doi.org/10.1007/s10958-008-9157-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9157-5

Keywords

Search

Navigation