# Commutators of trace zero matrices over principal ideal rings

## Abstract

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 *XY* − *YX* = *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.

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## References

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

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