The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring Mn(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.
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References
S. Akbari, M. Ghandehari, M. Hadian, and A. Mohammadian, “On commuting graphs of semisimple rings,” Linear Algebra Appl., 390, 345–355 (2004).
S. Akbari and A. Mohammadian, “On the zero-divisor graph of a commutative ring,” J. Algebra, 274, 847–855 (2004).
S. Akbari and A. Mohammadian, “Zero-divisor graphs of non-commutative rings.” J. Algebra, 296, 462–479 (2006).
S. Akbari, A. Mohammadian, H. Radjavi, and P. Raja, “On the diameters of commuting graphs,” Linear Algebra Appl., 418, 161–176 (2006).
D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring,” J. Algebra, 217, 434–447 (1999).
E. Artin, Geometric Algebra [Russian translation], Nauka, Moscow (1969).
B. R. Bakhadly, A. E. Guterman, and O. V. Markova, “Graphs defined by orthogonality,” Zap. Nauchn. Semin. POMI, 428, 49–80 (2014).
K. I. Beidar and A. V. Mikhalev, “Orthogonal completeness and algebraic systems,” Usp. Mat. Nauk, 40, No. 6, 79–115 (1985).
K. I. Beidar and A. V. Mikhalev, “The method of orthogonal completeness in the structure theory of rings,” Itogi Nauki Tekhniki, 4, 1–44 (1993).
F. Harary, Graph Theory [Russian translation], Mir, Moscow (1973).
I. N. Herstein, Noncommutative Rings [Russian translation], Mir, Moscow (1972).
A. E. Guterman and M. A. Efimov, “Monotone maps on matrices of index one,” Zap. Nauchn. Semin. POMI, 405, 67–96 (2012).
P. G. Ovchinnikov, “Automorphisms of the poset of skew projections,” J. Func. Anal., 115, 184–189 (1993).
P. Šemrl, “Order-preserving maps on the poset of idempotent matrices,” Acta Sci. Math. (Szeged), 69, 481–490 (2003).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 463, 2017, pp. 81–93.
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Guterman, A.E., Markova, O.V. Orthogonality Graphs of Matrices Over Skew Fields. J Math Sci 232, 797–804 (2018). https://doi.org/10.1007/s10958-018-3909-7
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DOI: https://doi.org/10.1007/s10958-018-3909-7