%0 Journal Article
%T Classification of rings with toroidal annihilating-ideal graph
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Krishnan, Selvakumar
%A P, Subbulakshmi
%D 2018
%\ 12/01/2018
%V 3
%N 2
%P 93-119
%! Classification of rings with toroidal annihilating-ideal graph
%K annihilating-ideal
%K planar
%K genus
%K local ring
%K annihilating-ideal graph
%R 10.22049/cco.2018.26060.1072
%X 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.
%U http://comb-opt.azaruniv.ac.ir/article_13745_64b16e21db453555d1fe39afe192d7e5.pdf