On the planarity of the -zero-divisor hypergraphs

Chelvam, T. Tamizh; Selvakumar, K.; Ramanathan, Varun

doi:10.1016/j.akcej.2015.11.011

Cited by 18 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Example 3. For R ∼ = Z 30 × Z 3 and I ∼ = 0 × Z 3 , we can separate Z I (R, 3) into 3 partite sets V 1 = (2, 0), (2, 1), (2, 2), (4, 0), (4, 1), (4, 2), (8, 0), (8,1), (8,2), (14, 0), (14,1), (14,2), (16,0), (16,1), (16,2), (22,0), (22,1), (22,2), (26, 0), (26, 1), (26, 2), (28, 0), (28, 1), (28, 2) , V 2 = (3, 0), (3,1), (3,2), (9, 0), (9, 1), (9,2), (21,0), (21,1), (21,2), (27, 0), (27, 1), (27,2)…”

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

k-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

Siriwong

Boonklurb

2021

Symmetry

View full text Add to dashboard Cite

Let R be a commutative ring with nonzero identity and k≥2 be a fixed integer. The k-zero-divisor hypergraph Hk(R) of R consists of the vertex set Z(R,k), the set of all k-zero-divisors of R, and the hyperedges of the form {a1,a2,a3,⋯,ak}, where a1,a2,a3,⋯,ak are k distinct elements in Z(R,k), which means (i) a1a2a3⋯ak=0 and (ii) the products of all elements of any (k−1) subsets of {a1,a2,a3,⋯,ak} are nonzero. This paper provides two commutative rings so that one of them induces a family of complete k-zero-divisor hypergraphs, while another induces a family of k-partite σ-zero-divisor hypergraphs, which illustrates unbalanced or asymmetric structure. Moreover, the diameter and the minimum length of all cycles or girth of the family of k-partite σ-zero-divisor hypergraphs are determined. In addition to a k-zero-divisor hypergraph, we provide the definition of an ideal-based k-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete k-partite k-uniform hypergraph, a complete k-uniform hypergraph, and a clique.

show abstract

Section: Proofmentioning

confidence: 99%

“…k-zero-divisor hypergraphs have been extensively studied. Eslahchi and Rahimi [7] studied a two-colorable H k (R), the connectedness and the completeness of three-zero-divisor hypergraphs; then Chelvam et al [8] studied the planarity of k-zero-divisor hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

k-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

Siriwong

Boonklurb

2021

Symmetry

View full text Add to dashboard Cite

show abstract

“…As a notation, the set of all k-zero divisor elements of R is denoted by Z k (R). By associating an hypergraph to R, some properties of Z k (R) are determined in [2,5,6]. Also, the 3-zero divisor elements of R/I has been studied as 3-zero divisor elements of R with respect to an ideal I, [4].…”

Section: Introductionmentioning

confidence: 99%

The 3-Zero Divisor and Quasi 3-Zero Divisor Elements of a Commutative Ring with Respect to an Ideal

Sevim

Babaei

Ulucak

2023

Preprint

View full text Add to dashboard Cite

For an ideal $I$ of a commutative ring $R$, we classify the set of all $3$-zero divisor elements of $R$ with respect to $I$. We prove that if $\bigcap_{i=1}^n P_i$ is a minimal prime decomposition of $\sqrt{I}$, then all elements of \begin{center} $\displaystyle{\bigcup_{i=1}^n P_i-\Big( \bigcup_{j=1}^n \Big\{x\in\bigcap_{i\in \{1,2,\ldots,n\}-\{j\}} P_i\backslash P_j: P_j\subseteq I:_Rx \Big\} \cup \sqrt{I}\Big)}$ \end{center} are $3$-zero divisor elements of $R$ with respect to $I$. Moreover, we determine $3$-zero divisors elements of $R$ with respect to $I$, when either $\sqrt{I}=I$ or $I$ is a primary ideal. Afterwards, we introduce quasi $3$-zero divisor element of $R$ with respect to $I$ and by using the minimal prime ideals of $R$ containing $I$ we characterize these elements. Moreover, we define quasi 3-zero divisor hypergraph of $R$ with respect to $I$ and we determine some properties of this hypergraph with the algebraic properties of $R$. 2000 Mathematics Subject Classification. 13A15, 13F30, 05C25.

show abstract

“…Later, researchers generalized the idea of a graph into a hypergraph. Chelvam et al [4] said that Eslahchi and Rahimi were the first who defined the notion of k-zero-divisor and its k-zero-divisor hypergraph. In there, Chelvam et al studied the planarity of k-zero-divisor hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

k-zero-divisor hypergraphs of finite commutative rings

Siriwong

View full text Add to dashboard Cite

Let R be a commutative ring with nonzero identity and k ≥ 2 be a fixed integer. The k-zero-divisor hypergraph Hk(R) of R consists of the vertex set Z(R; k), the set of all k-zero-divisors of R, and the (hyper)edges of the form {α1, α2, α3, .... αk} where α1, α2, α3, ..., αk are k distinct elements in Z(R; k), which means (i) α1α2α3..., αk = 0 and (ii) the products of all elements of any (k - 1)- subsets of {α1, α2, α3, ..., αk} are nonzero. This thesis provides (i) a necessary con- dition of commutative rings that implies the completeness of their k-zero-divisor hypergraphs; (ii) a necessary condition of commutative rings that implies the abil- ity to partition their set of all k-zero-divisors into k partite sets and the complete- ness of that k-partite k-zero-divisor hypergraphs; and (iii) a necessary condition of commutative rings that implies the ability to partition their set of all -zero- divisors into k partite sets, for some integer σ ≥ k. Moreover, the diameter and the minimum length of all cycles of those hypergraphs are determined.

show abstract

On the planarity of the -zero-divisor hypergraphs

Cited by 18 publications

References 7 publications

k-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

k-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

The 3-Zero Divisor and Quasi 3-Zero Divisor Elements of a Commutative Ring with Respect to an Ideal

k-zero-divisor hypergraphs of finite commutative rings

Contact Info

Product

Resources

About