Abstract
Abstract—In a paper published in 2008 P. A. Krylov showed that formal matrix rings Ks(R) and Kt(R) are isomorphic if and only if the elements s and t differ up to an automorphism by an invertible element. Similar dependence takes place in many cases. In this paper we consider formal matrix rings (and algebras) which have the same structure as incidence rings. We show that the isomorphism problem for formal matrix incidence rings can be reduced to the isomorphism problem for generalized incidence algebras. For these algebras, the direct assertion of Krylov’s theorem holds, but the converse is not true. In particular, we obtain a complete classification of isomorphisms of generalized incidence algebras of order 4 over a field. We also consider the isomorphism problem for special classes of formal matrix rings, namely, formal matrix rings with zero trace ideals.
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Krylov, P. A. “On the Isomorphism of Generalized Matrix Rings”, AlgebraLogic 47, No. 4, 258–262 (2008).
Anh, P. N. and van Wyk, L. “Automorphism Group of Generalized Triangular Matrix Rings”, Linear Algebra and Appl. 434, No. 4, 1018–1026 (2011).
Anh, P. N. and van Wyk, L. “Isomorphisms Between Strongly Triangular Matrix Rings”, Linear Algebra and Appl. 438, No. 11, 4374–4381 (2013).
Boboc, C., Dascalescu, S., and van Wyk, L. “Isomorphisms Between Morita Context Rings”, Linear and Multilinear Algebra 60, No. 5, 545–563 (2012).
Tapkin, D. T. “Formal Matrix Rings and a Generalization of Incidence Algebra”, Chebyshevsk. Sb. 16, No. 3, 422–449 (2015) [in Russian].
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Original Russian Text © D.T. Tapkin, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 12, pp. 84–91.
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Tapkin, D.T. Isomorphisms of formal matrix incidence rings. Russ Math. 61, 73–79 (2017). https://doi.org/10.3103/S1066369X1712009X
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DOI: https://doi.org/10.3103/S1066369X1712009X