Author

# Abbas Alilou

Bio: Abbas Alilou is an academic researcher from Azarbaijan Shahid Madani University. The author has contributed to research in topics: Commutative ring & Vertex (geometry). The author has an hindex of 3, co-authored 4 publications receiving 27 citations.

##### Papers

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TL;DR: In this article, the co-annihilating ideal graph of a commutative ring with identity was studied and its properties were investigated, where the vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever.

Abstract: Let be a commutative ring with identity. The co-annihilating-ideal graph of , denoted by , is a graph whose vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever . In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

19 citations

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TL;DR: In this article, the authors consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) =(0).

Abstract: Let R be a commutative ring with identity. In this paper, we consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we initiate the study of the graph ΩR∗ and we investigate its properties. In particular, we show that ΩR∗ is a connected graph with diam(ΩR∗) ≤ 3 unless R is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings R with at least two maximal ideals for which ΩR∗ are planar.

6 citations

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01 Dec 2016TL;DR: The study of the sum-annihilating essential ideal graph was initiated in this article, where it was shown that all rings whose essential ideal graphs are complete graphs are stars or complete graphs and sharp bounds on domination number of this graph were established.

Abstract: 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 of all non-zero annihilating ideals and twovertices $I$ and $J$ are adjacent whenever ${rm Ann}(I)+{rmAnn}(J)$ is an essential ideal. In this paper we initiate thestudy 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.

3 citations

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TL;DR: This paper initiates the study of the co-annihilating graph of a commutative ring and investigates its properties.

Abstract: Let R be a commutative ring with identity and 𝔘R be the set of all non-zero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜R, is a graph with vertex set 𝔘R and two vertices a and b are adjacent whenever Ann(a) ∩Ann(b) = (0). In this paper, we initiate the study of the co-annihilating graph of a commutative ring and we investigate its properties.

2 citations

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TL;DR: In this paper, the strong metric dimension of zero-divisor graphs is studied by transforming the problem of finding the vertex cover number of a strong resolving graph into a more well-known problem.

Abstract: In this paper, we study the strong metric dimension of zero-divisor graph $\Gamma(R)$ associated to a ring $R$. This is done by transforming the problem into a more well-known problem of finding the vertex cover number $\alpha(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring $\mathbb{Z}_n$ of integers modulo $n$ and the ring of Gaussian integers $\mathbb{Z}_n[i]$ modulo $n$. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.

9 citations

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TL;DR: The study of the essential ideal graph of a commutative ring with identity was initiated in this article, where the authors investigated its properties and showed that it is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal.

Abstract: Let R be a commutative ring with identity The essential ideal graph of R, denoted by ℰR, is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal In this paper, we initiate the study of the essential ideal graph of a commutative ring and we investigate its properties

8 citations

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12 Jun 2021TL;DR: A survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009 is given in this paper, where a simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with a ring R is considered.

Abstract: Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a non-zero intersection. The ideal structure of a ring reflects many ring theoretical properties. Thus much research has been conducted last few years to explore the properties of G(R). This is a survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009.

7 citations

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TL;DR: The commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one is characterized, whose vertex set is the set of all non-trivial ideals of R.

Abstract: Let R be a commutative ring with identity. We consider a simple graph associated with R, denoted by ΩR∗, whose vertex set is the set of all non-trivial ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we characterize the commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one.

5 citations

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TL;DR: In this article, the authors consider the problem of classifying rings R such that the complement of the annihilating ideal I of R is planar, i.e., the set of all annihilating ideals of R has a planar complement.

Abstract: The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. An ideal I of R is said to be an annihilating ideal if there exists $$r\in R\backslash \{0\}$$
such that $$Ir = (0)$$
. Let $$\mathbb {A}(R)$$
denote the set of all annihilating ideals of R and let us denote $$\mathbb {A}(R)\backslash \{(0)\}$$
by $$\mathbb {A}(R)^{*}$$
. Visweswaran and Patel (Discrete Math Algorithms Appl 6:22, 2014) introduced and studied a graph, denoted by $$\Omega (R)$$
, whose vertex set is $$\mathbb {A}(R)^{*}$$
and distinct vertices I, J are joined by an edge in this graph if and only if $$I + J\in \mathbb {A}(R)$$
. In Visweswaran and Sarman (Discrete Math Algorithms Appl 8:22, 2016), we investigated some properties of the complement of $$\Omega (R)$$
. The aim of this article is to classify rings R in order that $$\Omega (R)$$
be planar. We also consider the problem of classifying rings R such that the complement of $$\Omega (R)$$
is planar.

4 citations