Advertisement

Rings

  • Mark V. Sapir
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter, all rings are assumed to be associative. We start with a study of free associative algebras. This gives us the main syntactic tool to study rings. The first major result proved in this chapter is Shirshov’s height theorem, which is used to prove a theorem of Kaplansky. Then we prove the Dubnov–Ivanov–Nagata–Higman theorem about associative algebras satisfying the identity x n  = 0. Unlike semigroups, associative rings satisfying this identity are nilpotent. But there exist finitely generated non-nilpotent associative nil-algebras, and we describe the classical example of such an algebra constructed by Golod.

Keywords

Associative Algebra Homogeneous Polynomial Left Ideal Hilbert Series Associative Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 11.
    M. Aigner, G. Ziegler, Proofs from The Book, 4th edn. (Springer, Berlin/New York, 2010)CrossRefGoogle Scholar
  2. 13.
    A.S. Amitsur, J. Levitzki, Minimal identities for algebras. Proc. Am. Math. Soc. 1, 449–463 (1950)CrossRefMATHMathSciNetGoogle Scholar
  3. 16.
    R. Baer, Radical ideals. Am. J. Math. 65(4), 537–568 (1943)CrossRefMATHMathSciNetGoogle Scholar
  4. 29.
    A. Belov-Kanel, Some estimations for nilpotence of nil-algebras over a field of an arbitrary characteristic and height theorem. Commun. Algebra 20(10), 2919–2922 (1992)CrossRefGoogle Scholar
  5. 30.
    A. Belov-Kanel, On non-Specht varieties. Fundam. Prikl. Mat. 5(1), 47–66 (1999)MathSciNetGoogle Scholar
  6. 31.
    A. Belov-Kanel, Counterexamples to the Specht problem. Mat. Sb. 191(3), 13–24 (2000); trans. in Sb. Math. 191(3–4), 329–340 (2000)Google Scholar
  7. 32.
    A. Belov-Kanel, The Kurosh problem, the height theorem, the nilpotency of the radical, and the algebraicity identity. Fundam. Prikl. Mat. 13(2), 3–29 (2007); trans. in J. Math. Sci. (N. Y.) 154(2), 125–142 (2008)Google Scholar
  8. 33.
    A. Belov-Kanel, Local finite basis property and local representability of varieties of associative rings. Izv. Ross. Akad. Nauk Ser. Mat. 74(1), 3–134 (2010); trans. in Izv. Math. 74(1), 1–126 (2010)Google Scholar
  9. 34.
    A. Belov-Kanel, V. Borisenko, V. Latyshev, Monomial algebras (Algebra, 4). J. Math. Sci. (N. Y.) 87(3), 3463–3575 (1997)Google Scholar
  10. 35.
    A. Belov-Kanel, M.I. Kharitonov, Subexponential estimates in Shirshov’s theorem on height. Mat. Sb. 203(4), 81–102 (2012)CrossRefGoogle Scholar
  11. 36.
    A. Belov-Kanel, L.H. Rowen, Computational Aspects of Polynomial Identities. Volume 9 of Research Notes in Mathematics (A K Peters, Wellesley, 2005)Google Scholar
  12. 37.
    A. Belov-Kanel, L.H. Rowen, U. Vishne, Structure of Zariski-closed algebras. Trans. Am. Math. Soc. 362(9), 4695–4734 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 40.
    A. Belov-Kanel, L.H. Rowen, U. Vishne, PI-varieties associated to full quivers of representations of algebras. Trans. Am. Math. Soc. (to appear)Google Scholar
  14. 42.
    G.M. Bergman, Centralizers in free associative algebras. Trans. Am. Math. Soc. 137, 327–344 (1969)CrossRefMATHMathSciNetGoogle Scholar
  15. 50.
    A. Braun, The radical in a finitely generated P. I. algebra. Bull. Am. Math. Soc. (N.S.) 7, 385–386 (1982)Google Scholar
  16. 72.
    P.M. Cohn, Free Rings and Their Relations. London Mathematical Society Monographs, vol. 19, 2nd edn. (Academic/Harcourt Brace Jovanovich, London, 1985)Google Scholar
  17. 83.
    A. de Luca, S. Varricchio, Combinatorial properties of uniformly recurrent words and an application to semigroups. Int. J. Algebra Comput. 1(2), 227–246 (1991)CrossRefMATHGoogle Scholar
  18. 88.
    Dnestr notebook, Math. Inst. Siberian Acad. Sci., Novosibirsk, 1993Google Scholar
  19. 92.
    J. Dubnov, V. Ivanov, Sur l’abaissement du degré des polynômes en affineurs. C. R. (Doklady) Acad. Sci. URSS (N.S.) 41, 95–98 (1943)Google Scholar
  20. 112.
    E.S. Golod, On nil-algebras and finitely approximable p-groups. Izv. Akad. Nauk SSSR. Ser. Mat. 28, 273–276 (1964)MathSciNetGoogle Scholar
  21. 114.
    E.S. Golod, I.R. Shafarevich, On the class field tower. Izv. Akad. Nauk SSSR. Ser. Mat. 28, 261–272 (1964)MATHMathSciNetGoogle Scholar
  22. 124.
    A.V. Grishin, Examples of T-spaces and T-ideals of characteristic 2 without the finite basis property. Fundam. Prikl. Mat. 5(1), 101–118 (1999)MATHMathSciNetGoogle Scholar
  23. 125.
    A.V. Grishin, The variety of associative rings, which satisfy the identity x 32 = 0, is not Specht, in Formal Power Series and Algebraic Combinatorics Moscow, 2000 (Springer, Berlin/New York, 2000), pp. 686–691Google Scholar
  24. 140.
    C.K. Gupta, A.N. Krasilnikov, A simple example of a non-finitely based system of polynomial identities. Comm. Algebra 30(10), 4851–4866 (2002)CrossRefMATHMathSciNetGoogle Scholar
  25. 151.
    N. Herstein, Wedderburn’s theorem and a theorem of Jacobson. Am. Math. Mon. 68, 249–251 (1961)CrossRefMATHMathSciNetGoogle Scholar
  26. 152.
    G. Higman, On a conjecture of Nagata. Proc. Camb. Philos. Soc. 52, 1–4 (1956)CrossRefMATHMathSciNetGoogle Scholar
  27. 161.
    N. Jacobson, Structure of Rings (American Mathematical Society, Providence, 1956)MATHGoogle Scholar
  28. 164.
    T.J. Kaczynski, Another proof of Wedderburn’s theorem. Am. Math. Mon. 71(6), 652–653 (1964)CrossRefMATHMathSciNetGoogle Scholar
  29. 167.
    I. Kaplansky, Rings with a polynomial identity. Bull. Am. Math. Soc. 54, 575–580 (1948)CrossRefMATHMathSciNetGoogle Scholar
  30. 173.
    A.R. Kemer, Finite basability of identities of associative algebras. Algebra i Logika 26(5), 597–641 (1987)CrossRefMATHMathSciNetGoogle Scholar
  31. 174.
    A.R. Kemer, Ideals of Identities of Associative Algebras. Volume 82 of Translations of Mathematical Monographs (American Mathematical Society, Providence, 1991)Google Scholar
  32. 183.
    G. Köthe, Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollstandig irreduzibelist. Math. Z. 32, 161–186 (1930)CrossRefMATHMathSciNetGoogle Scholar
  33. 187.
    G.R. Krause, T.H. Lenagan, Growth of Algebras and Gelfand–Kirillov Dimension (Pitman, London, 1985)MATHGoogle Scholar
  34. 190.
    E.N. Kuzmin, On the Nagata–Higman theorem, in Mathematical Structures–Computational Mathematics–Mathematical Modeling, Proceedings Dedicated to the 16th Birthday of Academician L. Iliev, Sofia, 1975, pp. 101–107Google Scholar
  35. 192.
    V.N. Latyshev, Finite basis property of identities of certain rings. Usp. Mat. Nauk 32(4), 259–260 (1977)MATHGoogle Scholar
  36. 198.
    T.H. Lenagan, A. Smoktunowicz, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension. J. Am. Math. Soc. 20(4), 989–1001 (2007)CrossRefMATHMathSciNetGoogle Scholar
  37. 201.
    J. Levitzki, Prime ideals and the lower radical. Am. J. Math. 73(1), 25–29 (1951)CrossRefMATHMathSciNetGoogle Scholar
  38. 202.
    J. Lewin, Subrings of finite index in finitely-generated rings. J. Algebra, 5, 84–88 (1967)CrossRefMATHMathSciNetGoogle Scholar
  39. 205.
    A.A. Lopatin, On the nilpotency degree of the algebra with identity x n = 0. J. Algebra 371, 350–366 (2012)CrossRefMATHMathSciNetGoogle Scholar
  40. 214.
    J.H. Maclagan-Wedderburn. A theorem on finite algebras. Trans. Am. Math. Soc. 6, 349–352 (1905)CrossRefMATHMathSciNetGoogle Scholar
  41. 224.
    N.H. McCoy, Prime ideals in general rings. Am. J. Math. 71(4), 823–833 (1949)CrossRefMATHMathSciNetGoogle Scholar
  42. 272.
    Yu.P. Razmyslov, Identities of Algebras and their Representations (trans. from the 1989 Russian original by A.M. Shtern). Translations of Mathematical Monographs, vol. 138 (American Mathematical Society, Providence, 1994)Google Scholar
  43. 274.
    L.H. Rowen, Polynomial Identities in Ring Theory. Pure and Applied Mathematics, vol. 84 (Academic, [Harcourt Brace Jovanovich, Publishers], New York/London, 1980)Google Scholar
  44. 291.
    V.V. Shchigolev, Examples of infinitely based T-ideals. Fundam. Prikl. Mat. 5(1), 307–312 (1999)MATHMathSciNetGoogle Scholar
  45. 292.
    V.V. Shchigolev, Examples of infinitely based T-spaces. Mat. Sb. 191(3), 143–160 (2000); trans. in Sb. Math. 191(3–4), 459–476 (2000)Google Scholar
  46. 297.
    A.I. Shirshov, On some nonassociative nil-rings and algebraic algebras. Mat. Sb. 41(3), 381–394 (1957)MathSciNetGoogle Scholar
  47. 298.
    A.I. Shirshov, On rings with identity relations. Mat. Sb. 43(2), 277–283 (1957)MathSciNetGoogle Scholar
  48. 299.
    A.I. Shirshov, in Selected Works of A.I. Shirshov (trans. from the Russian by Murray Bremner and Mikhail V. Kotchetov), ed. by L.A. Bokut, V. Latyshev, I. Shestakov, E. Zel’manov. Contemporary Mathematicians (Birkhäuser, Basel, 2009)Google Scholar
  49. 305.
    A. Smoktunowicz, Some results in noncommutative ring theory, in International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006), pp. 259–269Google Scholar
  50. 306.
    W. Specht, Gesetze in Ringen. I. Math. Z. 52, 557–589 (1950)MATHMathSciNetGoogle Scholar
  51. 320.
    V.A. Ufnarovskii, On the use of graphs for calculating the basis, growth, and Hilbert series of associative algebras. Mat. Sb. 180(11), 1548–1560 (1989)Google Scholar
  52. 321.
    V.A. Ufnarovskij, Combinatorial and asymptotic methods in algebra, in Algebra, VI. Volume 57 of Encyclopaedia Mathematical Sciences (Springer, Berlin, 1995), pp. 1–196Google Scholar
  53. 336.
    E.I. Zel’manov, Some open problems in the theory of infinite dimensional algebras. J. Korean Math. Soc. 44(5), 1185–1195 (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mark V. Sapir
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations