Geodesic
$n$-Torsion clean and almost $n$-torsion clean matrix rings
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2021), pp. 52-63

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We (completely) determine those natural numbers $n$ for which the full matrix ring $\mathbb{M}_n(\mathbb{F}_2)$ and the triangular matrix ring $\mathbb{T}_n(\mathbb{F}_2)$ over the two elements field $\mathbb{F}_2$ 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 $n\times n$ ring $\mathbb{M}_n(\mathbb{F}_2)$ for all naturals $n\geq 1$, established in Linear Algebra Appl. (2013) by Breaz-Cǎlugǎreanu-Danchev-Micu and again in Linear Algebra Appl. (2018) by Šter as well as in Indag. Math. (2019) by Shitov.
Keywords: $n$-torsion clean ring, full matrix ring, triangular matrix ring, simple field.
Mots-clés : polynomial 
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     author = {A. C{\^\i}mpean and P. Danchev},
     title = {$n${-Torsion} clean and almost $n$-torsion clean matrix rings},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {52--63},
     publisher = {mathdoc},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2021_1_a3/}
}
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A. Cîmpean; P. Danchev. $n$-Torsion clean and almost $n$-torsion clean matrix rings. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2021), pp. 52-63. http://geodesic.mathdoc.fr/item/IVM_2021_1_a3/