On Zero-Divisor Graphs of <i>Z<sub>n</sub></i> When <i>n</i> Is Square-Free

On Zero-Divisor Graphs of <i>Z<sub>n</sub></i> When <i>n</i> Is Square-Free

In this article, some properties of the zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><msub><mi>Z</mi><mi>n</mi&g...

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Bibliographic Details
Main Authors:	Kholood Alnefaie, Nanggom Gammi, Saifur Rahman, Shakir Ali
Format:	Article
Language:	English
Published:	MDPI AG 2025-02-01
Series:	Axioms
Subjects:	graphs zero-divisor graphs bipartite graph k-partite graph congruence relation semisimple rings
Online Access:	https://www.mdpi.com/2075-1680/14/3/180
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Summary:	In this article, some properties of the zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><msub><mi>Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> are investigated when <i>n</i> is a square-free positive integer. It is shown that the zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><msub><mi>Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> of ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mi>n</mi></msub></semantics></math></inline-formula> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mn>2</mn><mi>k</mi></msup><mo>−</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>-partite graph when the prime decomposition of <i>n</i> contains <i>k</i> distinct square-free primes using the method of congruence relation. We present some examples, accompanied by graphic representations, to achieve the desired results. It is also obtained that the zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><msub><mi>Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is Eulerian if n is a square-free odd integer. Since <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mi>n</mi></msub></semantics></math></inline-formula> is a semisimple ring when <i>n</i> is square-free, the results can be generalized to characterize semisimple rings and modules, as well as rings satisfying Artinian and Noetherian conditions through the properties of their zero-divisor graphs. We endeavored to show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a partite graph with a certain condition on <i>n</i> and also that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a complete graph when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><msup><mi>p</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> for a prime <i>p</i> as part of a corollary. To prove these results, we employed the assistance of several theoretic congruence relations that grabbed our attention, making the investigation more interesting.
ISSN:	2075-1680