Alilou, A., Amjadi, J. (2016). The sum-annihilating essential ideal graph of a commutative ring. Communications in Combinatorics and Optimization, 1(2), 117-135. doi: 10.22049/cco.2016.13555

Abbas Alilou; Jafar Amjadi. "The sum-annihilating essential ideal graph of a commutative ring". Communications in Combinatorics and Optimization, 1, 2, 2016, 117-135. doi: 10.22049/cco.2016.13555

Alilou, A., Amjadi, J. (2016). 'The sum-annihilating essential ideal graph of a commutative ring', Communications in Combinatorics and Optimization, 1(2), pp. 117-135. doi: 10.22049/cco.2016.13555

Alilou, A., Amjadi, J. The sum-annihilating essential ideal graph of a commutative ring. Communications in Combinatorics and Optimization, 2016; 1(2): 117-135. doi: 10.22049/cco.2016.13555

The sum-annihilating essential ideal graph of a commutative ring

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 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.