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Year 2021, Volume: 50 Issue: 6, 1658 - 1666, 14.12.2021
https://doi.org/10.15672/hujms.822702

Abstract

References

  • [1] S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, The inclusion ideal graph of rings, Commun. Algebra 43 (6), 2457-2465, 2015.
  • [2] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217, 434-447, 1999.
  • [3] S. Aykaç, N. Akgüne and A.S. Çevik, Analysis of Zagreb indices over zero-divisor graphs of commutative rings, Asian-Eur. J. Math. 12 (6), 1-19, 2019. (Article ID 2040003)
  • [4] I. Beck, Coloring of commutative rings, J. Algebra 116, 208-226, 1988.
  • [5] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.
  • [6] L. Fuchs, On quasi-primary ideals, Acta. Sci. Math. (Szeged) 11 (3), 174-183, 1947.
  • [7] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math., 1992.
  • [8] G. Hahn and C. Tardif, Graph Homomorphisms: Structure and Symmetry, in: Graph Symmetry, 107-166, Springer, Dordrecht, 1997.
  • [9] I. Kaplansky, Commutative Rings (rev. ed.), University of Chicago Press, Chicago, 1974.
  • [10] D.A. Mojdeh and A.M. Rahimi, Dominating sets of some graphs associated to com- mutative rings, Commun. Algebra 40 (9), 3389-3396, 2012.
  • [11] N.J. Rad, S.H. Jafari and D.A. Mojdeh, On domination in zero-divisor graphs, Canad. Math. Bull. 56 (2), 407-411, 2013.
  • [12] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Commun. Algebra 31 (9), 4425-4443, 2003.

Ideal-based quasi zero divisor graph

Year 2021, Volume: 50 Issue: 6, 1658 - 1666, 14.12.2021
https://doi.org/10.15672/hujms.822702

Abstract

Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we introduce the ideal-based quasi zero divisor graph $Q\Gamma_{I}(R)$ of $R$ with respect to $I$ which is an undirected graph with vertex set $V=\{a\in R\backslash\sqrt{I}:$ $ab\in I$ for some $b\in R\backslash\sqrt{I}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab\in I$. We study the basic properties of this graph such as diameter, girth, dominaton number, etc. We also investigate the interplay between the ring theoretic properties of a Noetherian multiplication ring $R$ and the graph-theoretic properties of $Q\Gamma_{I}(R)$.

References

  • [1] S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, The inclusion ideal graph of rings, Commun. Algebra 43 (6), 2457-2465, 2015.
  • [2] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217, 434-447, 1999.
  • [3] S. Aykaç, N. Akgüne and A.S. Çevik, Analysis of Zagreb indices over zero-divisor graphs of commutative rings, Asian-Eur. J. Math. 12 (6), 1-19, 2019. (Article ID 2040003)
  • [4] I. Beck, Coloring of commutative rings, J. Algebra 116, 208-226, 1988.
  • [5] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.
  • [6] L. Fuchs, On quasi-primary ideals, Acta. Sci. Math. (Szeged) 11 (3), 174-183, 1947.
  • [7] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math., 1992.
  • [8] G. Hahn and C. Tardif, Graph Homomorphisms: Structure and Symmetry, in: Graph Symmetry, 107-166, Springer, Dordrecht, 1997.
  • [9] I. Kaplansky, Commutative Rings (rev. ed.), University of Chicago Press, Chicago, 1974.
  • [10] D.A. Mojdeh and A.M. Rahimi, Dominating sets of some graphs associated to com- mutative rings, Commun. Algebra 40 (9), 3389-3396, 2012.
  • [11] N.J. Rad, S.H. Jafari and D.A. Mojdeh, On domination in zero-divisor graphs, Canad. Math. Bull. 56 (2), 407-411, 2013.
  • [12] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Commun. Algebra 31 (9), 4425-4443, 2003.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ece Yetkin Çelikel 0000-0001-6194-656X

Angsuman Das 0000-0001-7647-4454

Cihat Abdioğlu 0000-0002-7874-2392

Publication Date December 14, 2021
Published in Issue Year 2021 Volume: 50 Issue: 6

Cite

APA Yetkin Çelikel, E., Das, A., & Abdioğlu, C. (2021). Ideal-based quasi zero divisor graph. Hacettepe Journal of Mathematics and Statistics, 50(6), 1658-1666. https://doi.org/10.15672/hujms.822702
AMA Yetkin Çelikel E, Das A, Abdioğlu C. Ideal-based quasi zero divisor graph. Hacettepe Journal of Mathematics and Statistics. December 2021;50(6):1658-1666. doi:10.15672/hujms.822702
Chicago Yetkin Çelikel, Ece, Angsuman Das, and Cihat Abdioğlu. “Ideal-Based Quasi Zero Divisor Graph”. Hacettepe Journal of Mathematics and Statistics 50, no. 6 (December 2021): 1658-66. https://doi.org/10.15672/hujms.822702.
EndNote Yetkin Çelikel E, Das A, Abdioğlu C (December 1, 2021) Ideal-based quasi zero divisor graph. Hacettepe Journal of Mathematics and Statistics 50 6 1658–1666.
IEEE E. Yetkin Çelikel, A. Das, and C. Abdioğlu, “Ideal-based quasi zero divisor graph”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 6, pp. 1658–1666, 2021, doi: 10.15672/hujms.822702.
ISNAD Yetkin Çelikel, Ece et al. “Ideal-Based Quasi Zero Divisor Graph”. Hacettepe Journal of Mathematics and Statistics 50/6 (December 2021), 1658-1666. https://doi.org/10.15672/hujms.822702.
JAMA Yetkin Çelikel E, Das A, Abdioğlu C. Ideal-based quasi zero divisor graph. Hacettepe Journal of Mathematics and Statistics. 2021;50:1658–1666.
MLA Yetkin Çelikel, Ece et al. “Ideal-Based Quasi Zero Divisor Graph”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 6, 2021, pp. 1658-66, doi:10.15672/hujms.822702.
Vancouver Yetkin Çelikel E, Das A, Abdioğlu C. Ideal-based quasi zero divisor graph. Hacettepe Journal of Mathematics and Statistics. 2021;50(6):1658-66.