Classification of rings with toroidal annihilating-ideal graph

Document Type : Original paper


1 Department of Mathematics Manonmaniam Sundaranar University Tirunelveli

2 Manonmaniam Sundaranar University


Let R be a non-domain commutative ring with identity and A(R) be the
set of non-zero ideals with non-zero annihilators. We call an ideal I of R, an
annihilating-ideal if there exists a non-zero ideal J of R such that IJ = (0).
The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex
set A(R) and two distinct vertices I and J are adjacent if and only if IJ =
(0). In this paper, we characterize all commutative Artinian nonlocal rings R
for which AG(R) has genus one.


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