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A survey on the Intersection graphs of ideals of rings

TL;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.
Citations
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Book ChapterDOI
25 Sep 2007

425 citations

Journal ArticleDOI
TL;DR: From the combination of knowledge and actions, someone can improve their skill and ability and this graph colorings tells you that any book will give certain knowledge to take all benefits.

134 citations

DOI
01 Dec 2018
TL;DR: In this paper, all commutative Artinian nonlocal rings with genus one have genus one and the annihilating-ideal graph is defined as the graph with the vertices of two distinct vertices that are adjacent if and only if their genus is genus one.
Abstract: Let $R$ be a non-domain commutative ring with identity and $A^*(R)$ be theset of non-zero ideals with non-zero annihilators. We call an ideal $I$ of $R$, anannihilating-ideal if there exists a non-zero ideal $J$ of $R$ such that $IJ = (0)$.The annihilating-ideal graph of $R$ is defined as the graph $AG(R)$ with the vertexset $A^*(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ =(0)$. In this paper, we characterize all commutative Artinian nonlocal rings $R$for which $AG(R)$ has genus one.

3 citations

DOI
01 Dec 2021
TL;DR: The annihilator-inclusion ideal graph of R, denoted by ξR, is a graph whose vertex set is the of all non-zero proper ideals of R, and two distinct vertices $I$ and $J$ are adjacent if and only if either ANN(I) ⊆ J or ANN(J)⊆ I.
Abstract: Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

2 citations

References
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Book
01 Jan 1969
TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

4,168 citations


"A survey on the Intersection graphs..." refers background in this paper

  • ...For details on terminology and results related to ring theory readers can refer [11, 46, 55, 60, 63, 67, 69] and for information on graph theory readers can refer [51, 68, 101]....

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Book
01 Jan 1998
TL;DR: Bounds on the domination number domination, independence and irredundance efficiency, redundancy and the duals changing and unchanging domination conditions on the dominating set varieties of domination multiproperty and multiset parameters sums and products of parameters dominating functions frameworks for domination domination complexity and algorithms are presented.
Abstract: Bounds on the domination number domination, independence and irredundance efficiency, redundancy and the duals changing and unchanging domination conditions on the dominating set varieties of domination multiproperty and multiset parameters sums and products of parameters dominating functions frameworks for domination domination complexity and algorithms.

3,265 citations


"A survey on the Intersection graphs..." refers background in this paper

  • ...One can refer [53] for various results related to domination parameters....

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Book
02 Nov 1977
TL;DR: Professor Massey's book developed from lecture notes of courses taught to Yale undergraduate and graduate students over a period of several years, and is the author of numerous research articles on algebraic topology and related topics.
Abstract: William S. Massey Professor Massey, born in Illinois in 1920, received his bachelor's degree from the University of Chicago and then served for four years in the U.S. Navy during World War II. After the War he received his Ph.D. from Princeton University and spent two additional years there as a post-doctoral research assistant. He then taught for ten years on the faculty of Brown University, and moved to his present position at Yale in 1960. He is the author of numerous research articles on algebraic topology and related topics. This book developed from lecture notes of courses taught to Yale undergraduate and graduate students over a period of several years.

1,215 citations


"A survey on the Intersection graphs..." refers background in this paper

  • ...We know that every non-orientable compact surface is a connected sum of finite copies of projective planes [70]....

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  • ...It is well known that any compact surface is either homeomorphic to a sphere or to a connected sum of g tori or to a connected sum of k projective planes [70]....

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Journal ArticleDOI
TL;DR: For each commutative ring R we associate a simple graph Γ(R) as discussed by the authors, and we investigate the interplay between the ring-theoretic properties of R and the graph-theory properties of Γ (R).

1,087 citations


"A survey on the Intersection graphs..." refers background in this paper

  • ...Many graphs related to the ring structure [33, 47] such as the zero-divisor graph [2, 3, 10, 81], the total graph and regular graph [1], the annihilating-ideal graph[14, 15], the annihilator-inclusion ideal graph [7], the toroidal annihilating-ideal graph [64], the sum-annihilating essential ideal graph [6] and the comaximal ideal graph [97]....

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Journal ArticleDOI

982 citations


Additional excerpts

  • ...It follows from Kuratowski’s Theorem [66] that a graph is planar if and only if it contains no subgraph homeomorphic from the complete graph K5 or from the complete bipartite graph K3,3....

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