Moreover, we state an equivalent condition for the chordality of the zero-divisor graph of an indecomposable ring and classify decomposable rings that have a chordal zero-divisor graph.
📌 3 Key Takeaways
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Objective & Hypothesis:We investigate when different graphs associated to commutative rings are chordal.
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Key Findings & Data:In particular, we characterize commutative rings $R$ with each of the following conditions: the total graph of $R$ is chordal; the total dot product or the zero-divisor dot product graph of $R$ is chordal; the comaximal graph of $R$ is chordal; $R$ is semilocal; and the unit graph or the Jacobson graph of $R$ is chordal.
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Conclusion & Impact:Moreover, we state an equivalent condition for the chordality of the zero-divisor graph of an indecomposable ring and classify decomposable rings that have a chordal zero-divisor graph.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Chordal graph · zero-divisor graph · total graph
Abstract
We investigate when different graphs associated to commutative rings are chordal. In particular, we characterize commutative rings $R$ with each of the following conditions: the total graph of $R$ is chordal; the total dot product or the zero-divisor dot product graph of $R$ is chordal; the comaximal graph of $R$ is chordal; $R$ is semilocal; and the unit graph or the Jacobson graph of $R$ is chordal. Moreover, we state an equivalent condition for the chordality of the zero-divisor graph of an indecomposable ring and classify decomposable rings that have a chordal zero-divisor graph.