The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo N

Yangjiang Wei (), Gaohua Tang () and Huadong Su
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Yangjiang Wei: Guangxi Teachers Education University
Gaohua Tang: Guangxi Teachers Education University
Huadong Su: Guangxi Teachers Education University

Indian Journal of Pure and Applied Mathematics, 2011, vol. 42, issue 5, 387-402

Abstract: 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.

Keywords: Commuting graph; quaternion algebra; connected component (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s13226-011-0025-5

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