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

Year 2017, Volume 21, Issue 21, 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, Volume 21, Issue 21, 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.

Details

Subjects Mathematics
Journal Section Articles
Authors

Jesse Gerald Smith Jr This is me

Publication Date January 17, 2017
Published in Issue Year 2017, Volume 21, Issue 21

Cite

Bibtex @research article { ieja296332, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2017}, volume = {21}, number = {21}, pages = {198 - 216}, doi = {10.24330/ieja.296332}, title = {WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED}, key = {cite}, author = {Smith Jr, Jesse Gerald} }
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 . DOI: 10.24330/ieja.296332
MLA Smith Jr, J. G. "WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED" . International Electronic Journal of Algebra 21 (2017 ): 198-216 <https://dergipark.org.tr/en/pub/ieja/issue/27921/296332>
Chicago Smith Jr, J. G. "WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED". International Electronic Journal of Algebra 21 (2017 ): 198-216
RIS TY - JOUR T1 - WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED AU - Jesse GeraldSmith Jr Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.296332 DO - 10.24330/ieja.296332 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 198 EP - 216 VL - 21 IS - 21 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.296332 UR - https://doi.org/10.24330/ieja.296332 Y2 - 2016 ER -
EndNote %0 International Electronic Journal of Algebra WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED %A Jesse Gerald Smith Jr %T WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 21 %N 21 %R doi: 10.24330/ieja.296332 %U 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
AMA Smith Jr J. G. WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED. IEJA. 2017; 21(21): 198-216.
Vancouver Smith Jr J. G. WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED. International Electronic Journal of Algebra. 2017; 21(21): 198-216.
IEEE J. G. Smith Jr , "WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED", International Electronic Journal of Algebra, vol. 21, no. 21, pp. 198-216, Jan. 2017, doi:10.24330/ieja.296332