%0 Journal Article
%T The sum-annihilating essential ideal graph of a commutative ring
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Alilou, Abbas
%A Amjadi, Jafar
%D 2016
%\ 12/01/2016
%V 1
%N 2
%P 117-135
%! The sum-annihilating essential ideal graph of a commutative ring
%K Commutative rings
%K annihilating ideal
%K essential ideal
%K genus of a graph
%R 10.22049/cco.2016.13555
%X Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$ is called an annihilating ideal if there exists $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of $R$ is called an essential ideal if $I$ has non-zero intersection with every other non-zero ideal of $R$. The sum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, is a graph whose vertex set is the set of all non-zero annihilating ideals and two vertices $I$ and $J$ are adjacent whenever ${rm Ann}(I)+{rm Ann}(J)$ is an essential ideal. In this paper we initiate the study of the sum-annihilating essential ideal graph. We first characterize all rings whose sum-annihilating essential ideal graph are stars or complete graphs and then establish sharp bounds on domination number of this graph. Furthermore determine all isomorphism classes of Artinian rings whose sum-annihilating essential ideal graph has genus zero or one.
%U http://comb-opt.azaruniv.ac.ir/article_13555_3f74eb186e2bee9fefcb8aa541b1f23c.pdf