<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: | , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/a77ff44c800b4d02b2f35042a25aa53b |
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| Sumario: | 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. |
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