Journal of Mathematical Sciences

, Volume 166, Issue 5, pp 675–681 | Cite as

Automorphisms and model-theory questions for Nilpotent matrix groups and rings

  • V. M. Levchuk
  • E. V. Minakova


Let R’ = NT(m, S). The purpose of this paper is to investigate elementary equivalences UT(n,K) ≡ UT(m, S) and Λ(R) ≡ Λ(R’) for arbitrary associative coefficient rings with identity. The main theorem gives the description of such equivalences for n > 4. In addition, we investigate isomorphisms and elementary equivalence of Jordan niltriangular matrix rings.


Associative Ring Matrix Ring Central Idempotent Adjoint Group Elementary Sentence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. V. Belegradek, “Model theory of unitriangular groups,” Am. Math. Soc. Transl., 195, No. 2, 1–116 (1999).MathSciNetGoogle Scholar
  2. 2.
    Yu. L. Ershov, “Elementary group theories,” Dokl. Akad. Nauk SSSR, 203, 1240–1243 (1972).MathSciNetGoogle Scholar
  3. 3.
    J. A. Gibbs, “Automorphisms of certain unipotent groups,” J. Algebra, 14, No. 2, 203–228 (1970).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    W. Hodges, Model Theory, Cambridge Univ. Press, Cambridge (1993).MATHCrossRefGoogle Scholar
  5. 5.
    H. J. Keisler and C. C. Chang, Model Theory [Russian translation], Mir, Moscow (1977).Google Scholar
  6. 6.
    F. Kuzucuoglu and V. M. Levchuk, “Isomorphisms of certain locally nilpotent finitary groups and associated rings,” Acta Appl. Math., 82, 169–181 (2004).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V. M. Levchuk, “Automorphisms of some nilpotent matrix groups and rings,” Dokl. Akad. Nauk SSSR, 16, No. 3, 756–760 (1975).MATHGoogle Scholar
  8. 8.
    V. M. Levchuk, “Connections between the unitriangular group and certain rings. II. Automorphisms,” Sib. Mat. Zh., 24, No. 4, 543–557 (1983).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. I. Malcev, “On a correspondence between rings and groups,” Mat. Sb., 50, 257–266 (1960).MathSciNetGoogle Scholar
  10. 10.
    A. I. Malcev, “The elementary properties of linear groups,” in: Certain Problems in Mathematics and Mechanics, Izd. SO AN SSSR, Novosibirsk (1961), pp. 110–132.Google Scholar
  11. 11.
    E. V. Minakova, “The elementary equivalence of Lie rings of niltriangular matrix rings over the commutative rings of coefficients,” Vestn. NGU, 8, No. 3, 100–104 (2008).Google Scholar
  12. 12.
    V. N. Remeslennikov and V. A. Roman’kov, “Model-theoretic and algorithmic questions of group theory,” in: Itogi Nauki Tekh., Ser. Algebra, Topol., Geom., Vol. 21, All-Union Institute for Scientific and Technical Information, Moscow (1983), pp. 3–79.Google Scholar
  13. 13.
    B. I. Rose, “The χ1-categoricity of strictly upper triangular matrix rings over algebraically closed fields,” J. Symb. Logic, 43, No. 2, 250–259 (1978).MATHCrossRefGoogle Scholar
  14. 14.
    C. R. Videla, “On the model theory of the ring NT(n,R),” J. Pure Appl. Algebra, 55, 289–302 (1988).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    C. R. Videla, “On the Mal’cev correspondence,” Proc. Am. Math. Soc., 109, 493–502 (1990).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    X. T. Wang, “Decomposition of Jordan automorphisms of strictly triangular matrix algebra over commutative rings,” Commun. Algebra, 35, 1133–1140 (2007).MATHCrossRefGoogle Scholar
  17. 17.
    X. T. Wang and H. You, “Decomposition of Jordan automorphisms of strictly triangular matrix algebra over local rings,” Linear Algebra Appl., 392, 183–193 (2004).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    W. H. Wheeler, “Model theory of strictly upper triangular matrix ring,” J. Symb. Logic, 45, 455–463 (1980).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Siberian Federal UniversityKrasnoyarskRussia

Personalised recommendations