A Generalization of the Zero-Divisor Graph for Modules

Safaeeyan, S.; Baziar, M.; Momtahan, Ehsan

doi:10.4134/jkms.2014.51.1.087

Cited by 12 publications

(7 citation statements)

References 14 publications

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“…Recently, the notions of zero divisor graph and annihilating-ideal graph have been extended from rings to modules in different ways. For instance, we can refer to [9] and [25]. In [9], the authors introduced and studied the annihilating-submodule graph.…”

Section: Introductionmentioning

confidence: 99%

On the Strongly Annihilating-Submodule Graph of a Module

Beyranvand

Farzi-safarabadi

2022

Hacettepe Journal of Mathematics and Statistics

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Section: Introductionmentioning

confidence: 99%

On the Strongly Annihilating-Submodule Graph of a Module

Beyranvand

Farzi-safarabadi

2022

Hacettepe Journal of Mathematics and Statistics

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“…The essential graph of R is a variation of the zero-divisor graph that changes the edge condition, and is introduced and studied in [10]. The essential graph of R is a simple undirected graph, denoted by EG(R), with vertex set Z * (R) and two distinct vertices x and y are adjacent if and only if Ann R (xy) is an essential ideal of R. Recently, a lot of research (e.g., [5,7,8,11,12]) has been devoted to the zerodivisor graph of a module (Definition 4.1). Let M be an R-module and let Z(M ) be its set of zero-divisors.…”

Section: Introductionmentioning

confidence: 99%

A Generalization of the Essential Graph for Modules Over Commutative Rings

Soheilnia

Payrovi

Behtoei

2021

International Electronic Journal of Algebra

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Let R be a commutative ring with nonzero identity and let M be a unitary R-module. The essential graph of M , denoted by EG(M ) is a simple undirected graph whose vertex set is Z(M )\Ann R (M ) and two distinct vertices x and y are adjacent if and only if Ann M (xy) is an essential submodule of M . Let r(Ann R (M )) = Ann R (M ). It is shown that EG(M ) is a connected graph with diam(EG(M )) ≤ 2. Whenever M is Noetherian, it is shown that EG(M ) is a complete graph if and only if either Z(M ) = r(Ann R (M )) or EG(M ) = K 2 and diam(EG(M )) = 2 if and only if there are x, y ∈ Z(M )\Ann R (M ) and p ∈ Ass R (M ) such that xy ∈ p. Moreover, it is proved that gr(EG(M )) ∈ {3, ∞}. Furthermore, for a Noetherian module M with r(Ann R (M )) = Ann R (M ) it is proved that |Ass R (M )| = 2 if and only if EG(M ) is a complete bipartite graph that is not a star.

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“…Recently, Kimball and LaGrange [13] generalized the definition to the idempotent divisor graph of a commutative ring. Besides this the zero divisor graph has also been extended to other algebraic structures like semi rings, Abelian groups, vector spaces, modules etc, (e.g., see the articles such as [5,8,9,23] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Metric and upper dimension of zero divisor graphs associated to commutative rings

Pirzada

Aijaz

2020

Acta Universitatis Sapientiae, Informatica

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Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

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A Generalization of the Zero-Divisor Graph for Modules

Abstract: Abstract. Let R be a commutative ring with identity and M an Rmodule. In this paper, we associate a graph to M , say Γ(M ), such that when M = R, Γ(M ) is exactly the classic zero-divisor graph.

Cited by 12 publications

References 14 publications

On the Strongly Annihilating-Submodule Graph of a Module

On the Strongly Annihilating-Submodule Graph of a Module

A Generalization of the Essential Graph for Modules Over Commutative Rings

Metric and upper dimension of zero divisor graphs associated to commutative rings

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