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Preprints Archive: Abstract of IC2010074 (2010)
Some results on the intersection graphs of ideals of rings
Document info: Pages 12, Figures 0.
Let $R$ be a ring with unity and $I(R)^*$ be the set of all non-trivial left ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is a graph with the vertex set $I(R)^*$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals are finite. Among other results, it is shown that for every ring, if the clique number of $G(R)$ is finite, then the chromatic number is finite too and if $R$ is a reduced ring both are equal.