<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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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/11/1980 |
| _version_ | 1797508386784280576 |
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| author | Pinkaew Siriwong Ratinan Boonklurb |
| author_facet | Pinkaew Siriwong Ratinan Boonklurb |
| author_sort | Pinkaew Siriwong |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:02:18Z |
| format | Article |
| id | doaj.art-a77ff44c800b4d02b2f35042a25aa53b |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T05:02:18Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-a77ff44c800b4d02b2f35042a25aa53b2023-11-23T01:42:48ZengMDPI AGSymmetry2073-89942021-10-011311198010.3390/sym13111980<i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative RingsPinkaew Siriwong0Ratinan Boonklurb1Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, ThailandDepartment of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, ThailandLet <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.https://www.mdpi.com/2073-8994/13/11/1980<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 hypergraph |
| spellingShingle | Pinkaew Siriwong Ratinan Boonklurb <i>k</i>-Zero-Divisor and Ideal-Based <i>k</i>-Zero-Divisor Hypergraphs of Some Commutative Rings Symmetry <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 |
| title | <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_short | <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 |
| 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 |
| url | https://www.mdpi.com/2073-8994/13/11/1980 |
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