Abstract
In the present paper, we study formal matrix rings over a given ring and determinants of such matrices.
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References
A. N. Abyzov and D. T. Tapkin, “On some classes of formal matrix rings,” Izv. Vyssh. Uchebn. Zaved., Mat., 59, No. 3, 1–12 (2015).
Y. Baba and K. Oshiro, Classical Artinian Rings and Related Topics, World Scientific, London (2009).
J.-J. Climent, P. R. Navarro, and L. Tortosa, “On the arithmetic of the endomorphism ring End(ℤ p × \( {\mathbb{Z}}_{p^2} \)),” Appl. Algebra Eng. Commun. Comput., 22, No. 2, 91–108 (2011).
M. Cohen, “Morita context related to finite automorphism groups of rings,” Pacific J. Math., 98, 37–54 (1982).
R. M. Fossum, A. Griffith, and I. Reiten, “Trivial extensions of Abelian categories,” in: Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory, Lect. Notes Math., Vol. 456, Springer, Berlin (1975), pp. 1–122.
O. Gurgun, S. Halicioglu, and A. Harmanci, Quasipolarity of Generalized Matrix Rings, arXiv: 1303.3173v1[math.RA] (2013).
O. Gurgun, S. Halicioglu, and A. Harmanci, Strong J-Cleanness of Formal Matrix Rings, arXiv: 1308.4105v1[math.RA] (2013).
R. A. Horn and Ch. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge (2012).
Q. Huang and G. Tang, “Quasipolar property of trivial Morita context,” Int. J. Pure Appl. Math., 90, No. 4, 423–431 (2014).
Q. Huang, G. Tang, and Y. Zhou, “Quasipolar property of generalized matrix rings,” Commun. Algebra, 42, No. 9, 3883–3894 (2014).
K. H. Kim, Boolean Matrix Theory and Applications, Marcel Dekker, New York (1982).
P. A. Krylov, “Isomorphisms of generalized matrix rings,” Algebra Logic, 47, No. 4, 258–262 (2008).
P. A. Krylov, “The group K 0 of a generalized matrix ring,” Algebra Logic, 52, No. 3, 250–261 (2013).
P. A. Krylov, “Calculation of the group K 1 of a generalized matrix ring,” Sib. Math. J., 55, No. 4, 639–644 (2014).
P. A. Krylov, “Determinants of generalized matrices of order 2,” J. Math. Sci., to appear.
P. A. Krylov, A. V. Mikhalev, and A. A. Tuganbaev, Endomorphism Rings of Abelian Groups, Kluwer Academic, Dordrecht (2003).
P. A. Krylov and A. A. Tuganbaev, “Modules over formal matrix rings,” J. Math. Sci., 171, No. 2, 248–295 (2010).
P. A. Krylov and A. A. Tuganbaev, Modules over Discrete Valuation Domains, De Gruyter Exp. Math., Vol. 43, Walter de Gruyter, Berlin (2008).
I. Palmer, “The global homological dimension of semitrivial extensions of rings,” Math. Scand., 37, 223–256 (1975).
A. D. Sands, “Radicals and Morita contexts,” J. Algebra, 24, 335–345 (1973).
G. Tang, Ch. Li, and Y. Zhou, “Study of Morita contexts,” Commun. Algebra, 42, 1668–1681 (2014).
G. Tang and Y. Zhou, “Strong cleanness of generalized matrix rings over a local ring,” Linear Algebra Appl., 437, 2546–2559 (2012).
G. Tang and Y. Zhou, “A class of formal matrix rings,” Linear Algebra Appl., 438, No. 12, 4672–4688 (2013).
A. Tuganbaev, Semidistributive Modules and Rings, Kluwer Academic, Dordrecht (1998).
A. Tuganbaev, Ring Theory. Arithmetical Modules and Rings [in Russian], MCCME, Moscow (2009).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 1, pp. 65–119, 2014.
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Krylov, P.A., Tuganbaev, A.A. Formal Matrices and Their Determinants. J Math Sci 211, 341–380 (2015). https://doi.org/10.1007/s10958-015-2610-3
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DOI: https://doi.org/10.1007/s10958-015-2610-3