Abstract. This article studies the zero divisor graphs of the ring of Lipschitz integers modulo n. In particular we focus on the number of vertices, the diameter and the girth. We also give some results regarding the domination number of these graphs.

In this paper we investigate the structure of a,b Z/nZ , the quaternion rings over Z/nZ. It is proved that these rings are isomor-otherwise. We also prove that the ring a,b Z/nZ is isomorphic to M2(Z/nZ) if and only if n is odd and that all quaternion algebras defined over Z/nZ are isomorphic if and only if n ≡ 0 (mod 4).Mathematics Subject Classification. 11R52, 16-99.

We prove that the diameter of the commuting graph of the full matrix ring over the real numbers is at most five. This answers, in the affirmative, a conjecture proposed by Akbari-Mohammadian-radjavi-Raja, for the special case of the field of real numbers.

For a division ring D, finite dimensional over its center F, we prove that the commuting graph Γ(M n (D)), where n ≥ 3, is connected if and only if the following condition is satisfied ( * )For any noncentral matrix A ∈ M n (D) if the algebra A is a field extension of F of degree nk, then there exists a proper intermediate field between F and A .Furthermore, if the graph Γ(M n (D)) is connected we prove that its diameter is between four and six.

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In this paper we study the isomorphisms of generalized Hamilton quaternions a,b R where R is a finite unital commutative ring of odd characteristic and a, b ∈ R. We obtain the number of non-isomorphic classes of generalized Hamilton quaternions in the case where R is a principal ideal ring.