Chen, Huanyin

  • Received : 2011.02.16
  • Published : 2012.05.31


An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. We characterize, in this article, the strongly nil cleanness of $2{\times}2$ and $3{\times}3$ matrices over $R[x]/(x^2-1)$ where $R$ is a commutative local ring with characteristic 2. Matrix decompositions over fields are derived as special cases.


strongly nil matrix;characteristic polynomial;local ring


  1. G. Borooah, A. J. Diesl, and T. J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (2008), no. 1, 281-296.
  2. H. Chen, On strongly J-clean rings, Comm. Algebra 38 (2010), no. 10, 3790-3804.
  3. H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, Hackensack, NJ: World Scientific, 2011.
  4. H. Chen, On uniquely clean rings, Comm. Algebra 39 (2011), no. 1, 189-198.
  5. A. J. Diesl, Classes of Strongly Clean Rings, Ph.D. Thesis, University of California, Berkeley, 2006.
  6. T. J. Dorsey, Cleanness and Strong Cleanness of Rings of Matrices, Ph.D. Thesis, University of California, Berkeley, 2006.
  7. L. Fan and X. Yang, A note on strongly clean matrix rings, Comm. Algebra 38 (2010), no. 3, 799-806.
  8. J. E. Humphreys, Introduction to Lie Algebra and Representation Theory, Springer- Verlag, Beijing, 2006.
  9. Y. Li, Strongly clean matrix rings over local rings, J. Algebra 312 (2007), no. 1, 397-404.
  10. W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra 27 (1999), no. 8, 3583-3592.
  11. W. K. Nicholson, Clean rings: a survey, Advances in Ring Theory, World Sci. Publ., Hackensack, NJ, 2005, 181-198.
  12. X. Yang and Y. Zhou, Some families of strongly clean rings, Linear Algebra Appl. 425 (2007), no. 1, 119-129.

Cited by

  1. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute vol.15, pp.1, 2017,
  2. Nil-quasipolar rings vol.20, pp.1, 2014,
  3. Strongly Clean Matrices Over Power Series vol.56, pp.2, 2016,