The sum-annihilating essential ideal graph of a commutative ring

Document Type: Original paper

Authors

Azarbaijan Shahid Madani University

Abstract

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.

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