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Characterization of $2\times 2$ nil-clean matrices over integral domains

Year 2018, Volume: 5 Issue: 3, 117 - 127, 08.10.2018
https://doi.org/10.13069/jacodesmath.451229

Abstract

Let $R$ be any ring with identity. An element $a \in R$ is called nil-clean, if $a=e+n$ where $e$ is an idempotent element and $n$ is a nil-potent element. In this paper we give necessary and sufficient conditions for a $2\times 2$ matrix over an integral domain $R$ to be nil-clean.

References

  • [1] D. Alpern, Generic Two Integer Variable Equation Solver, 2018, available at www.alpertron.com.ar/QUAD.HTM.
  • [2] T. Andreescu, D. Andrica, Quadratic Diophantine Equations, Springer, New York, 2015.
  • [3] D. Andrica, G. Calugareanu, A nil–clean 2x2 matrix over the integers which is not clean, J. Algebra Appl. 13(6) (2014) 1450009.
  • [4] D. K. Basnet, J. Bhattacharyya, Nil clean graph of rings, arXiv:1701.07630 [math.RA], https://arxiv.org/abs/1701.07630.
  • [5] A. T. Block Gorman, Generalizations of Nil Clean to Ideals, Wellesley College, Honors Thesis Collection, (388) 2016.
  • [6] S. Breaz, G. Calugareanu, P. Danchev, T. Micu, Nil–clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115–3119.
  • [7] A. J. Diesl, Nil–clean rings, J. Algebra 383(1) (2013) 197–211.
  • [8] Diophantine Equation ax + by + cz = d Solver, www.mathafou.free.fr/ex e_en/exedioph3.html.
  • [9] S. Hadjirezaei, S. Karimzadeh, On the nil–clean matrix over a UFD, J. Alg. Struc. Appl. 2(2) (2015) 49–55.
  • [10] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278.
  • [11] I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers, JohnWiley–Sons, 3rd edition, 1972.
  • [12] S. Sahinkaya, G. Tang, Y. Zhou, Nil–clean group rings, J. Algebra Appl. 16(7) (2017) 1750135.
  • [13] F. Smarandache, Existence and number of solutions of Diophantine quadratic equations with two unkowns in ZZ and IN, arXiv:0 704.3716 [math.GM], http://arxiv.org/abs/0704.3716.
Year 2018, Volume: 5 Issue: 3, 117 - 127, 08.10.2018
https://doi.org/10.13069/jacodesmath.451229

Abstract

References

  • [1] D. Alpern, Generic Two Integer Variable Equation Solver, 2018, available at www.alpertron.com.ar/QUAD.HTM.
  • [2] T. Andreescu, D. Andrica, Quadratic Diophantine Equations, Springer, New York, 2015.
  • [3] D. Andrica, G. Calugareanu, A nil–clean 2x2 matrix over the integers which is not clean, J. Algebra Appl. 13(6) (2014) 1450009.
  • [4] D. K. Basnet, J. Bhattacharyya, Nil clean graph of rings, arXiv:1701.07630 [math.RA], https://arxiv.org/abs/1701.07630.
  • [5] A. T. Block Gorman, Generalizations of Nil Clean to Ideals, Wellesley College, Honors Thesis Collection, (388) 2016.
  • [6] S. Breaz, G. Calugareanu, P. Danchev, T. Micu, Nil–clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115–3119.
  • [7] A. J. Diesl, Nil–clean rings, J. Algebra 383(1) (2013) 197–211.
  • [8] Diophantine Equation ax + by + cz = d Solver, www.mathafou.free.fr/ex e_en/exedioph3.html.
  • [9] S. Hadjirezaei, S. Karimzadeh, On the nil–clean matrix over a UFD, J. Alg. Struc. Appl. 2(2) (2015) 49–55.
  • [10] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278.
  • [11] I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers, JohnWiley–Sons, 3rd edition, 1972.
  • [12] S. Sahinkaya, G. Tang, Y. Zhou, Nil–clean group rings, J. Algebra Appl. 16(7) (2017) 1750135.
  • [13] F. Smarandache, Existence and number of solutions of Diophantine quadratic equations with two unkowns in ZZ and IN, arXiv:0 704.3716 [math.GM], http://arxiv.org/abs/0704.3716.
There are 13 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Kota Nagalakshmi Rajeswari This is me 0000-0002-7722-2491

Umesh Gupta This is me 0000-0001-7732-5768

Publication Date October 8, 2018
Published in Issue Year 2018 Volume: 5 Issue: 3

Cite

APA Rajeswari, K. N., & Gupta, U. (2018). Characterization of $2\times 2$ nil-clean matrices over integral domains. Journal of Algebra Combinatorics Discrete Structures and Applications, 5(3), 117-127. https://doi.org/10.13069/jacodesmath.451229
AMA Rajeswari KN, Gupta U. Characterization of $2\times 2$ nil-clean matrices over integral domains. Journal of Algebra Combinatorics Discrete Structures and Applications. October 2018;5(3):117-127. doi:10.13069/jacodesmath.451229
Chicago Rajeswari, Kota Nagalakshmi, and Umesh Gupta. “Characterization of $2\times 2$ Nil-Clean Matrices over Integral Domains”. Journal of Algebra Combinatorics Discrete Structures and Applications 5, no. 3 (October 2018): 117-27. https://doi.org/10.13069/jacodesmath.451229.
EndNote Rajeswari KN, Gupta U (October 1, 2018) Characterization of $2\times 2$ nil-clean matrices over integral domains. Journal of Algebra Combinatorics Discrete Structures and Applications 5 3 117–127.
IEEE K. N. Rajeswari and U. Gupta, “Characterization of $2\times 2$ nil-clean matrices over integral domains”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 5, no. 3, pp. 117–127, 2018, doi: 10.13069/jacodesmath.451229.
ISNAD Rajeswari, Kota Nagalakshmi - Gupta, Umesh. “Characterization of $2\times 2$ Nil-Clean Matrices over Integral Domains”. Journal of Algebra Combinatorics Discrete Structures and Applications 5/3 (October 2018), 117-127. https://doi.org/10.13069/jacodesmath.451229.
JAMA Rajeswari KN, Gupta U. Characterization of $2\times 2$ nil-clean matrices over integral domains. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018;5:117–127.
MLA Rajeswari, Kota Nagalakshmi and Umesh Gupta. “Characterization of $2\times 2$ Nil-Clean Matrices over Integral Domains”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 5, no. 3, 2018, pp. 117-2, doi:10.13069/jacodesmath.451229.
Vancouver Rajeswari KN, Gupta U. Characterization of $2\times 2$ nil-clean matrices over integral domains. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018;5(3):117-2.