Characterization of $2\times 2$ nil-clean matrices over integral domains
Keywords:
Nil-clean matrix, Idempotent matrix, Nil-potent matrix, Diophantine equation
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.
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D. Andrica, G. Calugareanu, A nil–clean 2x2 matrix over the integers which is not clean, J. Algebra Appl. 13(6) (2014) 1450009.
D. K. Basnet, J. Bhattacharyya, Nil clean graph of rings, arXiv:1701.07630 [math.RA], https://arxiv.org/abs/1701.07630.
A. T. Block Gorman, Generalizations of Nil Clean to Ideals, Wellesley College, Honors Thesis Collection, (388) 2016.
S. Breaz, G. Calugareanu, P. Danchev, T. Micu, Nil–clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115–3119.
A. J. Diesl, Nil–clean rings, J. Algebra 383(1) (2013) 197–211.
Diophantine Equation ax + by + cz = d Solver, www.mathafou.free.fr/ex e_en/exedioph3.html.
S. Hadjirezaei, S. Karimzadeh, On the nil–clean matrix over a UFD, J. Alg. Struc. Appl. 2(2) (2015) 49–55.
W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278.
I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers, JohnWiley–Sons, 3rd edition, 1972.
S. Sahinkaya, G. Tang, Y. Zhou, Nil–clean group rings, J. Algebra Appl. 16(7) (2017) 1750135.
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.
