Some results on the complement of the comaximal ideal graphs of commutative rings

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Abstract

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring. Recall from Ye and Wu (J Algebra Appl 11(6):1250114, 2012) that the comaximal ideal graph of R denoted by \({\mathscr {C}}(R)\) is an undirected graph whose vertex set is the set of all proper ideals I of R such that \(I\not \subseteq J(R)\), where J(R) is the Jacobson radical of R and distinct vertices IJ are joined by an edge in this graph if and only if \(I + J = R\). The aim of this article is to study the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \(({\mathscr {C}}(R))^{c}\), where \(({\mathscr {C}}(R))^{c}\) is the complement of the comaximal ideal graph of R.

Keywords

Complement of the comaximal ideal graph of a commutative ring Diameter Girth Clique number 

Mathematics Subject Classification

13A15 05C25 

Notes

Acknowledgements

We are very much thankful to the referee for many helpful suggestions and are very much thankful to Professor M. Fontana for his support.

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Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Department of MathematicsSaurashtra UniversityRajkotIndia

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