n-torsion clean and almost n-torsion clean matrix rings

2019/12/02 by Cimpean, Andrada, Danchev, Peter
#13B99 #16E50 #16U99 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1912.01104

Abstract

We completely determine those natural numbers n for which the full matrix ring M_n(F₂) and the triangular matrix ring T_n(F₂) over the two elements field F₂ are either n-torsion clean or are almost n-torsion clean, respectively. These results somewhat address and settle a question, recently posed by Danchev-Matczuk in Contemp. Math. (2019) as well as they supply in a more precise aspect the nil-cleanness property of the full matrix n× n ring M_n(F₂) for all naturals n ≥ 1, established in Linear Algebra and Appl. (2013) by Breaz-Calugareanu-Danchev-Micu and again in Linear Algebra & Appl. (2018) by Ster as well as in Indag. Math. (2020) by Shitov.

Related

• Rings and finite fields whose elements are sums or differences of tripotents and potents
• n-Torsion Clean Rings
• Very clean matrices over local rings
• Strong cleanness of the 2× 2 matrix ring over a general local ring
• Nil-clean matrix rings