The sum-annihilating essential ideal graph of a commutative‎ ‎ring

• Main Authors: A‎. ‎Alilou, J‎. ‎Amjadi
• Format: Article
• Language: English
• Published: Azarbaijan Shahide Madani University 2016-06-01
• Series: Communications in Combinatorics and Optimization
• Subjects: Commutative rings‎; ‎annihilating ideal‎; ‎essential ideal‎; ‎genus of a graph
• Online Access: http://comb-opt.azaruniv.ac.ir/article_13555_0.html

Let $R$ be a commutative ring with identity‎. ‎An ideal $I$ of a ring $R$‎ ‎is called an annihilating ideal if there exists $r\in R\setminus \{0\}$ 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 graphs are stars or complete graphs and then we establish sharp bounds on the domination number of this graph‎. ‎Furthermore‎, ‎we determine all {isomorphism classes} of Artinian rings whose sum-annihilating essential ideal graphs have genus zero or one‎.