Intersection graphs of ideals of rings

In this paper, we consider the intersection graph G(R) of nontrivial left ideals of a ring R. We characterize the rings R for which the graph G(R) is connected and obtain several necessary and sufficient conditions on a ring R such that G(R) is complete. For a commutative ring R with identity, we show that G(R) is complete if and only if G(R[x]) is also so. In particular, we determine the values of n for which G(Z(n)) is connected, complete, bipartite. planar or has a cycle. Next, we characterize finite graphs which arise as the intersection graphs of Z(n), and determine the set of all non-isomorphic graphs of Z(n) for a given number of vertices. We also determine the values of n for which the graph of Z(n) is Eulerian and Hamiltonian. (C) 2009 Published by Elsevier B.V.