Geodesic
On a class of 3-good formal matrix rings
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 67 (2020), pp. 55-62 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study formal matrix rings with values in a given ring and with a matrix of factors consisting of 0 and 1. Under the indicated restrictions, the formal matrix ring can be represented as a splitting extension of one of its nilpotent ideal using the product of ordinary matrix rings, and the question of the invertibility of the formal matrix is reduced to to the question of the invertibility of ordinary matrices over a ring. Under some additional conditions imposed on the matrix of factors, it is possible to use the well-known Henriksen theorem and prove that every element of a formal matrix ring is the sum of three invertible elements of this ring. Finally, we give examples of such formal matrix rings of orders 4 and 5.
Keywords: ring, good ring, formal matrix ring. 
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T. D. Norbosambuev; E. A. Timoshenko. On a class of 3-good formal matrix rings. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 67 (2020), pp. 55-62. http://geodesic.mathdoc.fr/item/VTGU_2020_67_a4/

[1] P. A. Krylov, T. D. Norbosambuev, “Automorphisms of formal matrix algebras”, Siberian Mathematical Journal, 59:5 (2018), 885–893 | DOI | DOI | MR | Zbl

[2] P. A. Krylov, T. D. Norbosambuev, “Group of automorphisms of one class of formal matrix algebras”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics, 2018, no. 53, 16–21 | DOI | DOI

[3] P. Vámos, “2-good rings”, Quarterly Journal of Mathematics, 56:3 (2005), 417–430 | DOI | MR | Zbl

[4] A. K. Srivastava, “A survey of rings generated by units”, Ann. Fac. Sci. Toulouse, Math., 19 (2010), 203–213 | DOI | MR | Zbl

[5] P. A. Krylov, A. A. Tuganbaev, “Formal matrices and their determinants”, Journal of Mathematical Sciences (New York), 211:3 (2015), 341–380 | DOI | MR | Zbl

[6] P. Krylov, A. Tuganbaev, Formal Matrices, Algebra and Applications, 23, Springer, 2017 | DOI | MR | Zbl

[7] P. A. Krylov, “Isomorphism of generalized matrix rings”, Algebra and Logic, 47:4 (2008), 258–262 | DOI | MR | Zbl

[8] K. Morita, “Duality for modules and its applications to the theory of rings with minimum condition”, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A, 6 (1958), 83–142 | MR | Zbl

[9] P. Loustaunau, J. Shapiro, “Morita contexts”, Non-Commutative Ring Theory, Lecture Notes in Mathematics, 1448, Springer, 1990, 80–92 | DOI | MR

[10] T. D. Norbosambuev, “On sums of diagonal and invertible formal matrices”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 4 (36), 34–40 | DOI

[11] T. D. Norbosambuev, “2-good diagonal formal matrices over the ring of integers”, All-Russia Youth Scientific Conference “Vse grani matematiki i mekhaniki”, sbornik statey, Izd. dom TGU, Tomsk, 2016, 6–12

[12] T. D. Norbosambuev, “Rank of a formal matrix. System of formal linear equations. Zero divisors”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics, 2018, no. 52, 5–12 | DOI | MR

[13] M. Henriksen, “Two classes of rings generated by their units”, J. Algebra, 31:1 (1974), 182–193 | DOI | MR | Zbl