The sum-annihilating essential ideal graph of a commutative ring
نویسندگان
چکیده
Let R be a commutative ring with identity. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R \ {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 AER, is a graph whose vertex set is the set of all non-zero annihilating ideals and two vertices I and J are adjacent whenever Ann(I) + 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.
منابع مشابه
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 $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of$R$ is called an essential ideal if $I$ has non-zero intersectionwith every other non-zero ideal of $R$. Thesum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, isa graph whose vertex set is the set...
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