<i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings

<i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> be a commutative ring with nonzero identity and <inlin...

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Autores principales: Pinkaew Siriwong, Ratinan Boonklurb
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
<i>k</i>-zero-divisor
<i>k</i>-zero-divisor hypergraph
ideal-based <i>k</i>-zero-divisor
ideal-based <i>k</i>-zero-divisor hypergraph
complete <i>k</i>-uniform hypergraph
<i>k</i>-partite <i>k</i>-uniform hypergraph
Mathematics
QA1-939
Acceso en línea:https://doaj.org/article/a77ff44c800b4d02b2f35042a25aa53b
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id oai:doaj.org-article:a77ff44c800b4d02b2f35042a25aa53b
record_format dspace
institution DOAJ
collection DOAJ
language EN
topic <i>k</i>-zero-divisor
<i>k</i>-zero-divisor hypergraph
ideal-based <i>k</i>-zero-divisor
ideal-based <i>k</i>-zero-divisor hypergraph
complete <i>k</i>-uniform hypergraph
<i>k</i>-partite <i>k</i>-uniform hypergraph
Mathematics
QA1-939
spellingShingle <i>k</i>-zero-divisor
<i>k</i>-zero-divisor hypergraph
ideal-based <i>k</i>-zero-divisor
ideal-based <i>k</i>-zero-divisor hypergraph
complete <i>k</i>-uniform hypergraph
<i>k</i>-partite <i>k</i>-uniform hypergraph
Mathematics
QA1-939
Pinkaew Siriwong
Ratinan Boonklurb
<i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings
description Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> be a commutative ring with nonzero identity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> be a fixed integer. The <i>k</i>-zero-divisor hypergraph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">H</mi><mi>k</mi></msub><mrow><mo>(</mo><mi mathvariant="script">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> consists of the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the set of all <i>k</i>-zero-divisors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>, and the hyperedges of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> are <i>k</i> distinct elements in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which means (i) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>2</mn></msub><msub><mi>a</mi><mn>3</mn></msub><mo>⋯</mo><msub><mi>a</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and (ii) the products of all elements of any (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>) subsets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> are nonzero. This paper provides two commutative rings so that one of them induces a family of complete <i>k</i>-zero-divisor hypergraphs, while another induces a family of <i>k</i>-partite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-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 <i>k</i>-partite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-zero-divisor hypergraphs are determined. In addition to a <i>k</i>-zero-divisor hypergraph, we provide the definition of an ideal-based <i>k</i>-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete <i>k</i>-partite <i>k</i>-uniform hypergraph, a complete <i>k</i>-uniform hypergraph, and a clique.
format article
author Pinkaew Siriwong
Ratinan Boonklurb
author_facet Pinkaew Siriwong
Ratinan Boonklurb
author_sort Pinkaew Siriwong
title <i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings
title_short <i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings
title_full <i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings
title_fullStr <i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings
title_full_unstemmed <i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings
title_sort <i>k</i>-zero-divisor and ideal-based <i>k</i>-zero-divisor hypergraphs of some commutative rings
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/a77ff44c800b4d02b2f35042a25aa53b
work_keys_str_mv AT pinkaewsiriwong ikizerodivisorandidealbasedikizerodivisorhypergraphsofsomecommutativerings
AT ratinanboonklurb ikizerodivisorandidealbasedikizerodivisorhypergraphsofsomecommutativerings
_version_ 1718410296841732096
spelling oai:doaj.org-article:a77ff44c800b4d02b2f35042a25aa53b2021-11-25T19:05:44Z<i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings10.3390/sym131119802073-8994https://doaj.org/article/a77ff44c800b4d02b2f35042a25aa53b2021-10-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/1980https://doaj.org/toc/2073-8994Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> be a commutative ring with nonzero identity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> be a fixed integer. The <i>k</i>-zero-divisor hypergraph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">H</mi><mi>k</mi></msub><mrow><mo>(</mo><mi mathvariant="script">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> consists of the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the set of all <i>k</i>-zero-divisors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>, and the hyperedges of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> are <i>k</i> distinct elements in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which means (i) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>2</mn></msub><msub><mi>a</mi><mn>3</mn></msub><mo>⋯</mo><msub><mi>a</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and (ii) the products of all elements of any (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>) subsets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> are nonzero. This paper provides two commutative rings so that one of them induces a family of complete <i>k</i>-zero-divisor hypergraphs, while another induces a family of <i>k</i>-partite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-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 <i>k</i>-partite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-zero-divisor hypergraphs are determined. In addition to a <i>k</i>-zero-divisor hypergraph, we provide the definition of an ideal-based <i>k</i>-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete <i>k</i>-partite <i>k</i>-uniform hypergraph, a complete <i>k</i>-uniform hypergraph, and a clique.Pinkaew SiriwongRatinan BoonklurbMDPI AGarticle<i>k</i>-zero-divisor<i>k</i>-zero-divisor hypergraphideal-based <i>k</i>-zero-divisorideal-based <i>k</i>-zero-divisor hypergraphcomplete <i>k</i>-uniform hypergraph<i>k</i>-partite <i>k</i>-uniform hypergraphMathematicsQA1-939ENSymmetry, Vol 13, Iss 1980, p 1980 (2021)

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