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WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED

Year 2017, , 198 - 216, 17.01.2017
https://doi.org/10.24330/ieja.296332

Abstract

Let R be a commutative ring with nonzero identity and I a proper
ideal of R. The ideal-based zero-divisor graph of R with respect to the ideal
I, denoted by ΓI (R), is the graph on vertices {x ∈ R \ I | xy ∈ I for some
y ∈ R\I}, where distinct vertices x and y are adjacent if and only if xy ∈ I. In
this paper, we give a complete classification of when an ideal-based zero-divisor
graph of a commutative ring is complemented or uniquely complemented based
on the total quotient ring of R/I.

References

  • [1] D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann
  • regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180(3) (2003), 221–241.
  • [2] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative
  • ring, J. Algebra, 217(2) (1999), 434–447.
  • [3] I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208–226.
  • [4] R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings,
  • Comm. Algebra, 30(2) (2002), 745–750.
  • [5] P. S. Livingston, Structure in Zero-Divisor Graphs of Commuative Rings, Masters
  • Thesis, Univeristy of Tennessee, Knoxville, TN, 1997
  • [6] S. P. Redmond, Generalizations of the Zero-Divisor Graph of a Ring, Ph.D.
  • Thesis, The University of Tennessee, Knoxville, TN, 2001.
  • [7] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring,
  • Comm. Algebra, 31(9) (2003), 4425–4443.
  • [8] J. G. Smith, Properties of Ideal-Based Zero-Divisor Graphs of Commutative
  • Rings, Ph.D. Thesis, The Univeristy of Tennessee, Knoxville, TN, 2014.
Year 2017, , 198 - 216, 17.01.2017
https://doi.org/10.24330/ieja.296332

Abstract

References

  • [1] D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann
  • regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180(3) (2003), 221–241.
  • [2] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative
  • ring, J. Algebra, 217(2) (1999), 434–447.
  • [3] I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208–226.
  • [4] R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings,
  • Comm. Algebra, 30(2) (2002), 745–750.
  • [5] P. S. Livingston, Structure in Zero-Divisor Graphs of Commuative Rings, Masters
  • Thesis, Univeristy of Tennessee, Knoxville, TN, 1997
  • [6] S. P. Redmond, Generalizations of the Zero-Divisor Graph of a Ring, Ph.D.
  • Thesis, The University of Tennessee, Knoxville, TN, 2001.
  • [7] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring,
  • Comm. Algebra, 31(9) (2003), 4425–4443.
  • [8] J. G. Smith, Properties of Ideal-Based Zero-Divisor Graphs of Commutative
  • Rings, Ph.D. Thesis, The Univeristy of Tennessee, Knoxville, TN, 2014.
There are 15 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Jesse Gerald Smith Jr This is me

Publication Date January 17, 2017
Published in Issue Year 2017

Cite

APA Smith Jr, J. G. (2017). WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED. International Electronic Journal of Algebra, 21(21), 198-216. https://doi.org/10.24330/ieja.296332
AMA Smith Jr JG. WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED. IEJA. January 2017;21(21):198-216. doi:10.24330/ieja.296332
Chicago Smith Jr, Jesse Gerald. “WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED”. International Electronic Journal of Algebra 21, no. 21 (January 2017): 198-216. https://doi.org/10.24330/ieja.296332.
EndNote Smith Jr JG (January 1, 2017) WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED. International Electronic Journal of Algebra 21 21 198–216.
IEEE J. G. Smith Jr, “WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED”, IEJA, vol. 21, no. 21, pp. 198–216, 2017, doi: 10.24330/ieja.296332.
ISNAD Smith Jr, Jesse Gerald. “WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED”. International Electronic Journal of Algebra 21/21 (January 2017), 198-216. https://doi.org/10.24330/ieja.296332.
JAMA Smith Jr JG. WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED. IEJA. 2017;21:198–216.
MLA Smith Jr, Jesse Gerald. “WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED”. International Electronic Journal of Algebra, vol. 21, no. 21, 2017, pp. 198-16, doi:10.24330/ieja.296332.
Vancouver Smith Jr JG. WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED. IEJA. 2017;21(21):198-216.