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The intersection graph G Z (Z n ) of zero-divisors of the ring Z n , the ring of integers modulo n is a simple undirected graph with the vertex set is Z(Z n ) * = Z(Z n ) \ {0}, the set of all nonzero zero-divisors of the ring Z n and for any two distinct vertices are adjacent if and only if their corresponding principal ideals have a nonzero intersection. We determine some results concerning the necessary and sufficient condition for the graph G Z (Z n ) is Hamiltonian. Also, we investigate for all values of for which the graph G Z (Z n ) is Hamiltonian and as an example we show that how the results give as easy proof of the existence of a Hamilton cycle.

In Kleinfeld [Assosymmetric rings, Proc. Amer. Math. Soc. 8 (1957) 983-986] showed that every commutator and every associator is in the nucleus N . In this paper, first we prove that the nucleus coincides with the center of the ring. Using this result and results of Kleinfeld, it is shown that a simple assosymmetric ring is third power associative and hence associative.

Let R be a commutative ring with Z(R), its set of zero divisors. The total zero divisor graph of R, denoted Z(Γ(R)) is the undirected (simple) graph with vertices Z(R) * =Z(R)-{0}, the set of nonzero zero divisors of R. and for distinct x, y z(R) * , the vertices x and y are adjacent if and only if x + y z(R). In this paper prove that let R is commutative ring such that Z (

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