Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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An Ideal-based Extended Zero-divisor Graph on Rings

  • Ashraf, Mohammad (Department of Mathematics, Aligarh Muslim University) ;
  • Kumar, Mohit (Department of Mathematics, Institute of Applied Sciences and Humanities, GLA University)
  • Received : 2019.06.10
  • Accepted : 2020.08.18
  • Published : 2022.09.30
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Abstract

Let R be a commutative ring with identity and let I be a proper ideal of R. In this paper, we study the ideal based extended zero-divisor graph 𝚪'I (R) and prove that 𝚪'I (R) is connected with diameter at most two and if 𝚪'I (R) contains a cycle, then girth is at most four girth at most four. Furthermore, we study affinity the connection between the ideal based extended zero-divisor graph 𝚪'I (R) and the ideal-based zero-divisor graph 𝚪I (R) associated with the ideal I of R. Among the other things, for a radical ideal of a ring R, we show that the ideal-based extended zero-divisor graph 𝚪'I (R) is identical to the ideal-based zero-divisor graph 𝚪I (R) if and only if R has exactly two minimal prime-ideals which contain I.

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Acknowledgement

The authors are indebted to the learned referee for several useful suggestions which have improved immensely.

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