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

Öz

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.

Anahtar Kelimeler

• Nil-clean matrix,Idempotent matrix,Nil-potent matrix,Diophantine equation

Kaynakça

1. [1] D. Alpern, Generic Two Integer Variable Equation Solver, 2018, available at www.alpertron.com.ar/QUAD.HTM.
2. [2] T. Andreescu, D. Andrica, Quadratic Diophantine Equations, Springer, New York, 2015.
3. [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. [4] D. K. Basnet, J. Bhattacharyya, Nil clean graph of rings, arXiv:1701.07630 [math.RA], https://arxiv.org/abs/1701.07630.
5. [5] A. T. Block Gorman, Generalizations of Nil Clean to Ideals, Wellesley College, Honors Thesis Collection, (388) 2016.
6. [6] S. Breaz, G. Calugareanu, P. Danchev, T. Micu, Nil–clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115–3119.
7. [7] A. J. Diesl, Nil–clean rings, J. Algebra 383(1) (2013) 197–211.
8. [8] Diophantine Equation ax + by + cz = d Solver, www.mathafou.free.fr/ex e_en/exedioph3.html.

9. [9] S. Hadjirezaei, S. Karimzadeh, On the nil–clean matrix over a UFD, J. Alg. Struc. Appl. 2(2) (2015) 49–55.
10. [10] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278.
11. [11] I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers, JohnWiley–Sons, 3rd edition, 1972.
12. [12] S. Sahinkaya, G. Tang, Y. Zhou, Nil–clean group rings, J. Algebra Appl. 16(7) (2017) 1750135.
13. [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.