The essential ideal graph of a commutative ring

Amjadi, Jafar

doi:10.1142/s1793557118500584

Cited by 9 publications

(5 citation statements)

References 6 publications

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“…Lemma 2.9. [2] Let S be a commutative ring with unity. If S contains no proper essential ideal, then J(S) = (0).…”

Section: Preliminariesmentioning

confidence: 99%

“…This was a novel and interesting approach in algebraic graph theory since the structure of an ideal is closely related to that of the corresponding ring. For more details one can refer [1,2,9,10,18]. M. Ye and T. Wu [18] set out to explore the comaximal ideal graph of a commutative ring C(S) in 2012.…”

Section: Introductionmentioning

confidence: 99%

“…In 2016, Azadi et al [6] investigated the perfection and planarity of the comaximal ideal graph. Lately, J. Amjadi [2] commenced the study of essential ideal graph E S , of a commutative ring S with unity. He establised the structural properties of E S with reference to the ring properties.…”

Section: Introductionmentioning

confidence: 99%

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Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Jamsheena¹,

Chithra²

2021

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Let S be a commutative ring with unity. The essential ideal graph of S, denoted by E S , is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices P and Q are adjacent whenever P + Q is an essential ideal. An essential ideal P of a ring S is an ideal P of S (P S), having nonzero intersection with every other ideal of S. The set M ax(S) contains all the maximal ideals of S. The Jacobson radical of S, J(S), is the set of intersection of all maximal ideals of S. The comaximal ideal graph of S, denoted by C(S), is a simple graph with vertices as proper ideals of A not contained in J(S) and the vertices P and Q are associated with an edge whenever P + Q = S. In this paper, we study the structural properties of the graph E S by using the ring theoretic concepts. We obtain a characterization for E S to be isomorphic to the comaximal ideal graph C(S). Moreover, we derive the structure theorem of E Zn and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of E Zn and determine the values of n for which E Zn is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to E Zn for some n.

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“…Lemma 2.9. [2] Let S be a commutative ring with unity. If S contains no proper essential ideal, then J(S) = (0).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Jamsheena¹,

Chithra²

2021

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“…Later in 2017, Wani and Pawar [18] studied essential ideals and weakly essential ideals in ternary semirings. Amjadi [2] studied the essential ideal graph of a commutative ring in 2018. In 2019, Murugadas et al [17] studied essential ideals in near-rings using k-quasi coincidence relation.…”

Section: Introductionmentioning

confidence: 99%

Essential Implicative UP-Filters and t-Essential Fuzzy Implicative UP-Filters of UP-Algebras

Gaketem

2023

Int. J. Anal. Appl.

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For the purpose of this research, we give the new concept of essential implicative UPfilter from the concept of essential UP-filters in UP-algebras. Although we know that the concept of implicative UP-filters is UP-filters, we have an example showing that the concept of essential UPfilters does not necessarily be essential implicative UP-filters. After that, we extend the concept of essential implicative UP-filters to t-essential fuzzy implicative UP-filters of UP-algebras and study their relationship based on characteristic functions and upper level subsets.

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“…Among possible graphs obtained from various algebraic structures, zero divisor graphs, annihilator graphs, and intersection graphs are the significant ones [5,6,12]. Amjadi [2], defined an essential ideal graph with respect to a commutative ring. The notion of the essential submodule and its dualizing concept namely, the superfluous submodule were studied by the authors (Anderson [3], Fluery [13]).…”

Section: Introductionmentioning

confidence: 99%

Graph with respect to superfluous elements in a lattice

Sahoo¹,

Panackal²,

Kedukodi³

et al. 2022

MMN

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We consider superfluous elements in a bounded lattice with 0 and 1, and introduce various types of graphs associated with these elements. The notions such as superfluous element graph (S(L)), join intersection graph (JI(L)) in a lattice, and in a distributive lattice, superfluous intersection graph (SI(L)) are defined. Dual atoms play an important role to find connections between the lattice-theoretic properties and those of corresponding graph-theoretic properties. Consequently, we derive some important equivalent conditions of graphs involving the cardinality of dual atoms in a lattice. We provide necessary illustrations and investigate properties such as diameter, girth, and cut vertex of these graphs.

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The essential ideal graph of a commutative ring

Cited by 9 publications

References 6 publications

Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Essential Implicative UP-Filters and t-Essential Fuzzy Implicative UP-Filters of UP-Algebras

Graph with respect to superfluous elements in a lattice

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