in this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. weobserve that over a commutative ring $r$, $bbb{ag}_*(_rm)$ isconnected and diam$bbb{ag}_*(_rm)leq 3$. moreover, if $bbb{ag}_*(_rm)$ contains a cycle, then $mbox{gr}bbb{ag}_*(_rm)leq 4$. also for an $r$-module $m$ with$bbb{a}_*(m)neq s(m)setminus {0}$, $...

In this paper we continue to study the strongly annihilating-submodule graph. addition providing more properties of graph, compare extensively graph with

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...

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...

In this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. Weobserve that over a commutative ring $R$, $Bbb{AG}_*(_RM)$ isconnected and diam$Bbb{AG}_*(_RM)leq 3$. Moreover, if $Bbb{AG}_*(_RM)$ contains a cycle, then $mbox{gr}Bbb{AG}_*(_RM)leq 4$. Also for an $R$-module $M$ with$Bbb{A}_*(M)neq S(M)setminus {0}$, $...

The rings considered in this article are commutative rings with identity $1neq 0$. The aim of this article is to define and study the exact annihilating-ideal graph of commutative rings. We discuss the interplay between the ring-theoretic properties of a ring and graph-theoretic properties of exact annihilating-ideal graph of the ring.

the rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. let $r$ be a ring. let $mathbb{a}(r)$ denote the set of all annihilating ideals of $r$ and let $mathbb{a}(r)^{*} = mathbb{a}(r)backslash {(0)}$. the annihilating-ideal graph of $r$, denoted by $mathbb{ag}(r)$  is an undirected simple graph whose vertex set is $mathbb{a}(r)...

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 i...

 The rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. Let $R$ be a ring. Let $mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}$. The annihilating-ideal graph of $R$, denoted by $mathbb{AG}(R)$  is an undirected simple graph whose vertex set is $mathbb{A}(R...