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Abstract
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Let R be a commutative ring with identity. We consider a simple graph associated with
R, denoted by Ω∗
R, whose vertex set is the set of all non-trivial ideals of R and two
distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In
this paper, we characterize the commutative Artinian non-local ring R for which Ω∗
R has
genus one and crosscap one.
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