Commutators of trace zero matrices over principal ideal rings

  • Alexander Stasinski


We prove that for every trace zero square matrix A of size at least 3 over a principal ideal ring R, there exist trace zero matrices X, Y over R such that XYYX = A. Moreover, we show that X can be taken to be regular mod every maximal ideal of R. This strengthens our earlier result that A is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is simpler than the earlier one.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. A. Albert and B. Muckenhoupt, On matrices of trace zero, Michigan Mathematical Journal 4 (1957), 1–3.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. A. Hermida and T. Sánchez-Giralda, Linear equations over commutative rings and determinantal ideals, Journal of Algebra 99 (1986), 72–79.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    T. J. Laffey and R. Reams, Integral similarity and commutators of integral matrices, Linear Algebra and its Applications 197/198 (1994), 671–689.Google Scholar
  4. [4]
    D. Lissner, Matrices over polynomial rings, Transactions of the American Mathematical Society 98 (1961), 285–305.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C. de Seguins Pazzis, Commutators from a hyperplane of matrices, Electronic Journal of Linear Algebra 27 (2014), 39–54.MathSciNetzbMATHGoogle Scholar
  6. [6]
    A. Stasinski, Similarity and commutators of matrices over principal ideal rings, Transactions of the American Mathematical Society 368 (2016), 2333–2354.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    R. C. Thompson, Matrices with zero trace, Israel Journal of Mathematics 4 (1966), 33–42.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    L. N. Vaserstein, Noncommutative number theory, in Algebraic K-Theory and Algebraic Number Theory, Contemporary Mathematics, Vol. 83, American Mathematical Society, Providence, RI, 1989, pp. 445–449.MathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham University South RdDurhamUK

Personalised recommendations