THE COMMUTING GRAPHS OF SOME SUBSETS IN THE QUATERNION ALGEBRA OVER THE RING OF INTEGERS MODULO N

Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s is an element of S vertical bar sx = xs for every x is an element of S}. The commuting graph of S, denoted by Gamma(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 Gamma(I(n)) and Gamma(N(n)). Moreover, it is proved that for n >= 2, Gamma(I(n)) is connected if and only if n has at least two odd prime factors, while Gamma(N(n)) is connected if and only if n not equal 2, 2(2), p,2p for all odd primes p.