• Chen, Huanyin (Department of Mathematics Hangzhou Normal University)
  • Received : 2009.12.08
  • Published : 2011.07.31


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. A ring is strongly nil clean in case each of its elements is strongly nil clean. We investigate, in this article, the strongly nil cleanness of 2${\times}$2 matrices over local rings. For commutative local rings, we characterize strongly nil cleanness in terms of solvability of quadratic equations. The strongly nil cleanness of a single triangular matrix is studied as well.


  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, Separative ideals, clean elements and unit-regularity, Comm. Algebra 34 (2006), no. 3, 911-921.
  3. H. Chen, Clean matrices over commutative rings, Czechoslovak Math. J. 59(134) (2009), no. 1, 145-158.
  4. H. Chen, On strongly J-clean rings, Comm. Algebra 38 (2011), no. 10, 3790-3804.
  5. W. Chen, A question on strongly clean rings, Comm. Algebra 34 (2006), no. 7, 2347-2350.
  6. A. J. Diesl, Classes of Strongly Clean Rings, Ph D. Thesis, University of California, Berkeley, 2006.
  7. J. E. Humphreys, Introduction to Lie Algebra and Representation Theory, Springer-Verlag, Beijing, 2006.
  8. W. K. Nicholson, Clean rings: a survey, Advances in ring theory, 181-198, World Sci. Publ., Hackensack, NJ, 2005.
  9. X. Yang and Y. Zhou, Some families of strongly clean rings, Linear Algebra Appl. 425 (2007), no. 1, 119-129.
  10. X. Yang and Y. Zhou, Strongly cleanness of the $2\times2$ matrix ring over a general local ring, J. Algebra 320 (2008), no. 6, 2280-2290.

Cited by

  1. Quasipolar Subrings of 3 x 3 Matrix Rings vol.21, pp.3, 2013,
  2. Nil clean rings vol.383, 2013,
  3. Nil-quasipolar rings vol.20, pp.1, 2014,