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On two extensions of the annihilating-ideal graph of commutative rings

  • Mohd Nazim ORCID logo EMAIL logo , Nadeem ur Rehman ORCID logo and Junaid Nisar ORCID logo
Published/Copyright: July 25, 2023
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 30 Issue 6

Abstract

Let R be a commutative ring with A ⁢ ( R ) its set of annihilating-ideals. The extended annihilating-ideal graph of R, denoted by AG ¯ ⁢ ( R ) , is an undirected graph with vertex set A ⁢ ( R ) * = A ⁢ ( R ) ∖ { 0 } and two vertices I 1 and I 2 are adjacent if and only if I 1 m ⁢ I 2 n = 0 with I 1 m ≠ 0 and I 2 n ≠ 0 , for some positive integers m and n. In this paper, we first study some basic properties of AG ¯ ⁢ ( R ) and then we investigate the relationship between the extended annihilating-ideal graph AG ¯ ⁢ ( R ) , the annihilator-ideal graph A I ⁢ ( R ) and the annihilating-ideal graph AG ⁢ ( R ) of a commutative ring R.

Keywords: Annihilating-ideal graph; commutative ring; extended annihilating-ideal graph; zero-divisor graph
MSC 2020: 13A15; 05C12; 05C25

Acknowledgements

The authors are deeply grateful to the referee for careful reading of the paper and helpful suggestions.

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Received: 2022-10-18
Revised: 2023-03-20
Accepted: 2023-03-29
Published Online: 2023-07-25
Published in Print: 2023-12-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/gmj-2023-2044
Keywords for this article
Annihilating-ideal graph; commutative ring; extended annihilating-ideal graph; zero-divisor graph

Articles in the same Issue

  1. Frontmatter
  2. On left and right Browder elements in Banach algebra relative to a bounded homomorphism
  3. Uniform excess of g-frames
  4. On normal partial isometries
  5. Gauss--Newton--Kurchatov method for the solution of non-linear least-square problems using ω-condition
  6. Maps preserving the bi-skew Jordan product on factor von Neumann algebras
  7. Boundary contact problems with regard to friction of couple-stress viscoelasticity for inhomogeneous anisotropic bodies (quasi-static cases)
  8. On uniform statistical convergence
  9. Approximate solution of the optimal control problem for non-linear differential inclusion on the semi-axes
  10. Energy-based localization of positive solutions for stationary Kirchhoff-type equations and systems
  11. e-reversibility of rings via quasinilpotents
  12. Weighted generalized Moore–Penrose inverse
  13. On two extensions of the annihilating-ideal graph of commutative rings
  14. Umbral treatment and lacunary generating function for Hermite polynomials
  15. A generalization of the canonical commutation relation and Heisenberg uncertainty principle for the orbital operators
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