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Matrix rings over a principal ideal domain in which elements are nil-clean

Year 2016, , 91 - 96, 15.05.2016
https://doi.org/10.13069/jacodesmath.82415

Abstract

An element of a ring $R$ is called nil-clean if it is the sum of an idempotent and a nilpotent element. A ring is called nil-clean if each of its elements is nil-clean. S. Breaz et al. in \cite{Bre} proved their main result that the matrix ring $\mathbb{M}_{ n}(F)$ over a field $F$ is nil-clean if and only if $F\cong \mathbb{F}_2$, where $\mathbb{F}_2$ is the field of two elements. M. T. Ko\c{s}an et al. generalized this result to a division ring. In this paper, we show that the $n\times n$ matrix ring over a principal ideal domain $R$ is a nil-clean ring if and only if $R$ is isomorphic to $\mathbb{F}_2$. Also, we show that the same result is true for the $2\times 2$ matrix ring over an integral domain $R$. As a consequence, we show that for a commutative ring $R$, if $\mathbb{M}_{ 2}(R)$ is a nil-clean ring, then dim$R=0$ and char${R}/{J(R)}=2$.

References

  • [1] S. Breaz, G. Calugareanu, P. Danchev, T. Micu, Nil-clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115-3119.
  • [2] J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, Comm. Algebra. 34(10) (2006) 3659–3674.
  • [3] A. J. Diesl, Classes of strongly clean rings, Ph. D. thesis, University of California, Berkeley, 2006.
  • [4] A. J. Diesl, Nil clean rings, J. Algebra. 383 (2013) 197–211.
  • [5] T. W. Hungerford, Algebra, Springer-Verlag, 1980.
  • [6] M.T. Kosan, T. K. Lee, Y. Zhou, When is every matrix over a division ring a sum of an idempotent and a nilpotent?, Linear Algebra Appl. 450 (2014) 7–12.
  • [7] T. Kosan, Z. Wang, Y. Zhou, Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra. 220(2) (2016) 633–646.
  • [8] W. K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra. 27(8) (1999) 3583–3592.
  • [9] G. Song, X. Guo, Diagonability of idempotent matrices over noncommutative rings, Linear Algebra Appl. 297(1-3) (1999) 1–7.
Year 2016, , 91 - 96, 15.05.2016
https://doi.org/10.13069/jacodesmath.82415

Abstract

References

  • [1] S. Breaz, G. Calugareanu, P. Danchev, T. Micu, Nil-clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115-3119.
  • [2] J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, Comm. Algebra. 34(10) (2006) 3659–3674.
  • [3] A. J. Diesl, Classes of strongly clean rings, Ph. D. thesis, University of California, Berkeley, 2006.
  • [4] A. J. Diesl, Nil clean rings, J. Algebra. 383 (2013) 197–211.
  • [5] T. W. Hungerford, Algebra, Springer-Verlag, 1980.
  • [6] M.T. Kosan, T. K. Lee, Y. Zhou, When is every matrix over a division ring a sum of an idempotent and a nilpotent?, Linear Algebra Appl. 450 (2014) 7–12.
  • [7] T. Kosan, Z. Wang, Y. Zhou, Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra. 220(2) (2016) 633–646.
  • [8] W. K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra. 27(8) (1999) 3583–3592.
  • [9] G. Song, X. Guo, Diagonability of idempotent matrices over noncommutative rings, Linear Algebra Appl. 297(1-3) (1999) 1–7.
There are 9 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Somayeh Hadjirezaei This is me

Somayeh Karimzadeh This is me

Publication Date May 15, 2016
Published in Issue Year 2016

Cite

APA Hadjirezaei, S., & Karimzadeh, S. (2016). Matrix rings over a principal ideal domain in which elements are nil-clean. Journal of Algebra Combinatorics Discrete Structures and Applications, 3(2), 91-96. https://doi.org/10.13069/jacodesmath.82415
AMA Hadjirezaei S, Karimzadeh S. Matrix rings over a principal ideal domain in which elements are nil-clean. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2016;3(2):91-96. doi:10.13069/jacodesmath.82415
Chicago Hadjirezaei, Somayeh, and Somayeh Karimzadeh. “Matrix Rings over a Principal Ideal Domain in Which Elements Are Nil-Clean”. Journal of Algebra Combinatorics Discrete Structures and Applications 3, no. 2 (May 2016): 91-96. https://doi.org/10.13069/jacodesmath.82415.
EndNote Hadjirezaei S, Karimzadeh S (May 1, 2016) Matrix rings over a principal ideal domain in which elements are nil-clean. Journal of Algebra Combinatorics Discrete Structures and Applications 3 2 91–96.
IEEE S. Hadjirezaei and S. Karimzadeh, “Matrix rings over a principal ideal domain in which elements are nil-clean”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 2, pp. 91–96, 2016, doi: 10.13069/jacodesmath.82415.
ISNAD Hadjirezaei, Somayeh - Karimzadeh, Somayeh. “Matrix Rings over a Principal Ideal Domain in Which Elements Are Nil-Clean”. Journal of Algebra Combinatorics Discrete Structures and Applications 3/2 (May 2016), 91-96. https://doi.org/10.13069/jacodesmath.82415.
JAMA Hadjirezaei S, Karimzadeh S. Matrix rings over a principal ideal domain in which elements are nil-clean. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3:91–96.
MLA Hadjirezaei, Somayeh and Somayeh Karimzadeh. “Matrix Rings over a Principal Ideal Domain in Which Elements Are Nil-Clean”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 2, 2016, pp. 91-96, doi:10.13069/jacodesmath.82415.
Vancouver Hadjirezaei S, Karimzadeh S. Matrix rings over a principal ideal domain in which elements are nil-clean. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3(2):91-6.