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 Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, Number 4(36), Pages 34–40 (Mi vtgu469)

MATHEMATICS

On sums of diagonal and invertible formal matrices

T. D. Norbosambuev

Tomsk State University, Tomsk, Russian Federation

Abstract: This paper concerns properties of $k$-good formal matrix rings $K_n$ of order $n$ with rings $R_1, R_2, …, R_n$ on the main diagonal and $R_i-R_j$-bimodules $M_{ij}$ on other places. In the ring theory, various matrix rings play an important role. Above all I mean formal matrix rings. Formal matrix rings generalize a notion of matrix ring of order $n$ over a given ring. Every ring with nontrivial idempotents is isomorphic to some formal matrix ring. The endomorphism ring of a decomposable module also is a formal matrix ring. The studies of such rings are quite useful for solving some problems on endomorphism rings of Abelian groups. In this paper I show that every matrix form $K_n$ is the sum of diagonal matrix and invertible matrix. Also I give one condition when $K_n$ is the $k$-good ring.

Keywords: ring, generalized matrix, formal matrix, $k$-good ring.

DOI: https://doi.org/10.17223/19988621/36/4

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UDC: 512.552+512.643.8

Citation: T. D. Norbosambuev, “On sums of diagonal and invertible formal matrices”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, no. 4(36), 34–40

Citation in format AMSBIB
\Bibitem{Nor15} \by T.~D.~Norbosambuev \paper On sums of diagonal and invertible formal matrices \jour Vestn. Tomsk. Gos. Univ. Mat. Mekh. \yr 2015 \issue 4(36) \pages 34--40 \mathnet{http://mi.mathnet.ru/vtgu469} \crossref{https://doi.org/10.17223/19988621/36/4} \elib{https://elibrary.ru/item.asp?id=24132030} 

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This publication is cited in the following articles:
1. Ts. D. Norbosambuev, “Rang formalnoi matritsy. Sistema formalnykh lineinykh uravnenii. Deliteli nulya”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2018, no. 52, 5–12
2. Ts. D. Norbosambuev, E. A. Timoshenko, “Ob odnom klasse 3-khoroshikh kolets formalnykh matrits”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 67, 55–62
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