Co-maximal graph of non-commutative rings

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Abstract

Let R be a ring (not necessarily commutative) with 1. Following Sharma and Bhatwadekar [P.K. Sharma, S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995) 124-127], we define a graph on R, Γ (R), with vertices as elements of non-units of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. In this paper, we investigate the behavior of Γ (R). We are able to prove that if R is left Artinian then Γ (R) - J (R) is connected and if Γ (R) - J (R) is a forest then Γ (R) - J (R) is a star graph, where J (R) is the Jacobson radical of R. For any finite field Fq, we obtain the minimal degree, the maximal degree, the connectivity, the clique number and the chromatic number of Γ (Mn (Fq)). Finally, for any finite field and any integer n ≥ 2, we prove that if R is a ring with identity and Γ (R) ≅ Γ (Mn (F)), then R ≅ Mn (F). We also prove that if R and S are two finite commutative rings with identity, and n, m ≥ 2 such that Γ (Mn (R)) ≅ Γ (Mm (S)), then n = m and R ≅ S provided that R is reduced. © 2008 Elsevier Inc. All rights reserved.

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Wang, H. J. (2009). Co-maximal graph of non-commutative rings. Linear Algebra and Its Applications, 430(2–3), 633–641. https://doi.org/10.1016/j.laa.2008.08.026

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