Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph

Anderson, David F.; LaGrange, John D.

doi:10.1016/j.jpaa.2011.12.002

Cited by 53 publications

(33 citation statements)

References 15 publications

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“…The compressed zero-divisor graph Γ E (R) is the (undirected) graph whose vertices are the equivalence classes induced by ∼ other than [0] and [1], such that distinct vertices [s] and [t] are adjacent in Γ E (R) if and only if st = 0. This graph was later studied more extensively in [3,8,17]. Now, we extend this definition to noncommutative rings.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Automorphisms of the Zero-Divisor Graph Over 2 × 2 Matrices

Ma¹,

Wang²,

Zhou³

2016

Journal of the Korean Mathematical Society

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Section: Preliminaries and Notationsmentioning

confidence: 99%

Automorphisms of the Zero-Divisor Graph Over 2 × 2 Matrices

Ma¹,

Wang²,

Zhou³

2016

Journal of the Korean Mathematical Society

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“…The following theorem reduces chordality of Γ(R) to chordality of Γ E (R) , the compressed zero-divisor graph of R defined in [17] and further studied in [3].…”

Section: Lemma 310mentioning

confidence: 99%

Chordality of graphs associated to commutative rings

Nikseresht¹

2018

Turk J Math

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We investigate when different graphs associated to commutative rings are chordal. In particular, we characterize commutative rings R with each of the following conditions: the total graph of R is chordal; the total dot product or the zero-divisor dot product graph of R is chordal; the comaximal graph of R is chordal; R is semilocal; and the unit graph or the Jacobson graph of R is chordal. Moreover, we state an equivalent condition for the chordality of the zero-divisor graph of an indecomposable ring and classify decomposable rings that have a chordal zero-divisor graph.

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“…The zero-divisor graph of R, denoted by Γ(R), is a graph with vertices Z(R) * = Z(R) \ {0} and two distinct vertices a and b are adjacent if and only if ab = 0, see [4,6]. The compressed zero-divisor graph of R, Γ E (R), that is constructed from equivalence classes of zero-divisors, rather than individual zero-divisors themselves was introduced in [11] and studied in some literatures, for examples [3,8,13]. This graph 578 S.B.…”

Section: Introductionmentioning

confidence: 99%

The intersection graph of annihilator submodules of a module

Pejman¹,

Payrovi²,

Babaei³

2019

Opuscula Math.

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Let R be a commutative ring and M be a Noetherian R-module. The intersection graph of annihilator submodules of M , denoted by GA(M) is an undirected simple graph whose vertices are the classes of elements of ZR(M) \ AnnR(M), for a, b ∈ R two distinct classes [a] and [b] are adjacent if and only if AnnM (a) ∩ AnnM (b) = 0. In this paper, we study diameter and girth of GA(M) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that GA(M) is complete if and only if ZR(M) is an ideal of R. Also, we show that if M is a finitely generated R-module with r(AnnR(M)) = AnnR(M) and |m − AssR(M)| = 1 and GA(M) is a star graph, then r(AnnR(M)) is not a prime ideal of R and |V (GA(M))| = | Min AssR(M)| + 1.

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Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph

Cited by 53 publications

References 15 publications

Automorphisms of the Zero-Divisor Graph Over 2 × 2 Matrices

Automorphisms of the Zero-Divisor Graph Over 2 × 2 Matrices

Chordality of graphs associated to commutative rings

The intersection graph of annihilator submodules of a module

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