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Abstract
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Let R be a commutative ring with identity. In this paper, we consider a simple graph
associated with R denoted by Ω∗R, whose vertex set is the set of all nonzero proper
ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or
IAnn(J) = (0). In this paper, we initiate the study of the graph Ω∗R and we investigate
its properties. In particular, we show that Ω∗R is a connected graph with diam(Ω∗R) ≤ 3
unless R is isomorphic to a direct product of two fields. Moreover, we characterize all
commutative rings R with at least two maximal ideals for which Ω∗R are planar.
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