Abstract
Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s ∈ S | sx = xs for every x ∈ S}. The commuting graph of S, denoted by Γ(S), is the graph with vertex set S \ Z(S) such that two different vertices x and y are adjacent if and only if xy = yx. In this paper, let I n , N n be the sets of all idempotents, nilpotent elements in the quaternion algebra ℤ n [i, j, k], respectively. We completely determine Γ(I n ) and Γ(N n ). Moreover, it is proved that for n ≥ 2, Γ(I n ) is connected if and only if n has at least two odd prime factors, while Γ(N n ) is connected if and only if n ∈ 2, 22, p, 2p for all odd primes p.
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References
David F. Anderson and Ayman Badawi, The total graph of a commutative ring, J. of Algebra, 320 (2008), 2706–2719.
David F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. of Algebra, 217 (1999), 434–447.
S. Akbari, M. Ghandehari, M. Hadian and A. Mohammadian, On commuting graphs of semisimple rings, Linear Algebra Appl., 390 (2004), 345–355.
S. Akbari and P. Raja, Commuting graphs of some subsets in simple rings, Linear Algebra Appl., 416 (2006), 1038–1047.
T. Y. Lam, A first course in noncommutative rings, Springer Verlag, New York, 1991.
C. D. Pan and C. B. Pan, Elementary number theory (second edition), Beijing university publishing company, Beijing, 2005.
Y. J. Wei and G. H. Tang, The spectrum and radicals of quaternion algebra ℤn[ixxx jxxx k], J. Guangxi Teachers Education University, 26(1) (2009), 1–10.
Y. J. Wei and G. H. Tang, The zero-divisors and unit group of quaternion algebra ℤn[ixxx jxxx k], Guangxi Sciences, 16(2) (2009), 147–150.
Y. J. Wei, G. H. Tang and H. D. Su, The commuting graph of the quaternion algebra over residue classes of integers, Ars Combinatoria, 95 (2010), 113–127.
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Wei, Y., Tang, G. & Su, H. The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo N . Indian J Pure Appl Math 42, 387–402 (2011). https://doi.org/10.1007/s13226-011-0025-5
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DOI: https://doi.org/10.1007/s13226-011-0025-5