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Year 2018, , 176 - 202, 11.01.2018
https://doi.org/10.24330/ieja.373663

Abstract

References

  • D. D. Anderson, Commutative rngs, in Multiplicative Ideal Theory in Commutative Algebra: A tribute to the work of Robert Gilmer (J. W. Brewer et al., Eds.), Springer-Verlag, New York, (2006), 1-20.
  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
  • D. D. Anderson and J. Stickles, Commutative rings with nitely generated multiplicative semigroup, Semigroup Forum, 60(3) (2000), 436-443.
  • D. F. Anderson, On the diameter and girth of a zero-divisor graph, II, Houston J. Math., 34(2) (2008), 361-371.
  • D. F. Anderson, M. C. Axtell and J. A. Stickles, Jr., Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives (M. Fontana et al., Eds.), Springer-Verlag, New York, (2011), 23-45.
  • D. F. Anderson and A. Badawi, The zero-divisor graph of a commutative semi- group: A survey, in Groups, Modules, and Model Theory-Surveys and Recent Developments, In Memory of Rudiger Gobel (M. Droste et al., Eds.), Springer- Verlag, Cham, (2017), 23-39.
  • D. F. Anderson and J. D. LaGrange, Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph, J. Pure Appl. Algebra, 216(7) (2012), 1626-1636.
  • D. F. Anderson and J. D. LaGrange, Some remarks on the compressed zero- divisor graph, J. Algebra, 447 (2016), 297-321.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • 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.
  • D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210(2) (2007), 543-550.
  • M. Axtell, J. Stickles and W. Trambachls, Zero-divisor ideals and realizable zero-divisor graphs, Involve, 2(1) (2009), 17-27.
  • M. Axtell, J. Stickles and J. Warfel, Zero-divisor graphs of direct products of commutative rings, Houston J. Math, 32(4) (2006), 985-994.
  • R. Ballieu, Anneaux nis; systemes hypercomplexes de rang trois sur un corps commtatif, Annales de la Societe Scienti c de Bruxelles, Serie I, 61 (1947), 222-227.
  • R. Ballieu, Anneaux nis a module de type (p; p2), Annales de la Societe Scienti c de Bruxelles, Serie I, 63 (1949), 11-23.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • B. Bollaboas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979.
  • J. Coykendall, S. Sather-Wagsta , L. Sheppardson and S. Spiro , On zero divisor graphs, in Progress in Commutative Algebra II: Closures, Finiteness and Factorization (C. Francisco et al., Eds.), Walter de Gruyter, Berlin, (2012), 241-299.
  • F. DeMeyer and L. DeMeyer, Zero divisor graphs of semigroups, J. Algebra, 283(1) (2005), 190-198.
  • F. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum, 65(2) (2002), 206-214.
  • R. Gilmer and J. Mott, Associative rings of order p3, Proc. Japan Acad., 49 (1973), 795-799.
  • T. G. Lucas, The diameter of a zero-divisor graph, J. Algebra, 301(1) (2006), 174-193.
  • S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30(7) (2002), 3533-3558.
  • S. P. Redmond, On zero-divisor graphs of small nite commutative rings, Discrete Math., 307(9-10) (2007), 1155-1166.
  • S. P. Redmond, Corrigendum to: \On zero-divisor graphs of small nite com- mutative rings" [Discrete Math., 307 (2007), 1155-1166], Discrete Math., 307(21) (2007), 2449-2452.
  • S. P. Redmond, Counting zero-divisors, in Commutative Rings: New Research, Nova Sci. Publ., Hauppauge, NY, (2009), 7-12.
  • S. Spiro and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39(7) (2011), 2338-2348.
  • D. Weber, Various Topics on Graphical Structures Placed on Commutative Rings, PhD dissertation, The University of Tennessee, 2017.

The zero-divisor graph of a commutative ring without identity

Year 2018, , 176 - 202, 11.01.2018
https://doi.org/10.24330/ieja.373663

Abstract

Let R be a commutative ring. The zero-divisor graph of R is the
(simple) graph 􀀀(R) with vertices the nonzero zero-divisors of R, and two
distinct vertices x and y are adjacent if and only if xy = 0. In this article, we
investigate 􀀀(R) when R does not have an identity, and we determine all such
zero-divisor graphs with 14 or fewer vertices.

References

  • D. D. Anderson, Commutative rngs, in Multiplicative Ideal Theory in Commutative Algebra: A tribute to the work of Robert Gilmer (J. W. Brewer et al., Eds.), Springer-Verlag, New York, (2006), 1-20.
  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
  • D. D. Anderson and J. Stickles, Commutative rings with nitely generated multiplicative semigroup, Semigroup Forum, 60(3) (2000), 436-443.
  • D. F. Anderson, On the diameter and girth of a zero-divisor graph, II, Houston J. Math., 34(2) (2008), 361-371.
  • D. F. Anderson, M. C. Axtell and J. A. Stickles, Jr., Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives (M. Fontana et al., Eds.), Springer-Verlag, New York, (2011), 23-45.
  • D. F. Anderson and A. Badawi, The zero-divisor graph of a commutative semi- group: A survey, in Groups, Modules, and Model Theory-Surveys and Recent Developments, In Memory of Rudiger Gobel (M. Droste et al., Eds.), Springer- Verlag, Cham, (2017), 23-39.
  • D. F. Anderson and J. D. LaGrange, Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph, J. Pure Appl. Algebra, 216(7) (2012), 1626-1636.
  • D. F. Anderson and J. D. LaGrange, Some remarks on the compressed zero- divisor graph, J. Algebra, 447 (2016), 297-321.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • 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.
  • D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210(2) (2007), 543-550.
  • M. Axtell, J. Stickles and W. Trambachls, Zero-divisor ideals and realizable zero-divisor graphs, Involve, 2(1) (2009), 17-27.
  • M. Axtell, J. Stickles and J. Warfel, Zero-divisor graphs of direct products of commutative rings, Houston J. Math, 32(4) (2006), 985-994.
  • R. Ballieu, Anneaux nis; systemes hypercomplexes de rang trois sur un corps commtatif, Annales de la Societe Scienti c de Bruxelles, Serie I, 61 (1947), 222-227.
  • R. Ballieu, Anneaux nis a module de type (p; p2), Annales de la Societe Scienti c de Bruxelles, Serie I, 63 (1949), 11-23.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • B. Bollaboas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979.
  • J. Coykendall, S. Sather-Wagsta , L. Sheppardson and S. Spiro , On zero divisor graphs, in Progress in Commutative Algebra II: Closures, Finiteness and Factorization (C. Francisco et al., Eds.), Walter de Gruyter, Berlin, (2012), 241-299.
  • F. DeMeyer and L. DeMeyer, Zero divisor graphs of semigroups, J. Algebra, 283(1) (2005), 190-198.
  • F. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum, 65(2) (2002), 206-214.
  • R. Gilmer and J. Mott, Associative rings of order p3, Proc. Japan Acad., 49 (1973), 795-799.
  • T. G. Lucas, The diameter of a zero-divisor graph, J. Algebra, 301(1) (2006), 174-193.
  • S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30(7) (2002), 3533-3558.
  • S. P. Redmond, On zero-divisor graphs of small nite commutative rings, Discrete Math., 307(9-10) (2007), 1155-1166.
  • S. P. Redmond, Corrigendum to: \On zero-divisor graphs of small nite com- mutative rings" [Discrete Math., 307 (2007), 1155-1166], Discrete Math., 307(21) (2007), 2449-2452.
  • S. P. Redmond, Counting zero-divisors, in Commutative Rings: New Research, Nova Sci. Publ., Hauppauge, NY, (2009), 7-12.
  • S. Spiro and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39(7) (2011), 2338-2348.
  • D. Weber, Various Topics on Graphical Structures Placed on Commutative Rings, PhD dissertation, The University of Tennessee, 2017.
There are 28 citations in total.

Details

Journal Section Articles
Authors

David F. Anderson

Darrin Weber This is me

Publication Date January 11, 2018
Published in Issue Year 2018

Cite

APA Anderson, D. F., & Weber, D. (2018). The zero-divisor graph of a commutative ring without identity. International Electronic Journal of Algebra, 23(23), 176-202. https://doi.org/10.24330/ieja.373663
AMA Anderson DF, Weber D. The zero-divisor graph of a commutative ring without identity. IEJA. January 2018;23(23):176-202. doi:10.24330/ieja.373663
Chicago Anderson, David F., and Darrin Weber. “The Zero-Divisor Graph of a Commutative Ring Without Identity”. International Electronic Journal of Algebra 23, no. 23 (January 2018): 176-202. https://doi.org/10.24330/ieja.373663.
EndNote Anderson DF, Weber D (January 1, 2018) The zero-divisor graph of a commutative ring without identity. International Electronic Journal of Algebra 23 23 176–202.
IEEE D. F. Anderson and D. Weber, “The zero-divisor graph of a commutative ring without identity”, IEJA, vol. 23, no. 23, pp. 176–202, 2018, doi: 10.24330/ieja.373663.
ISNAD Anderson, David F. - Weber, Darrin. “The Zero-Divisor Graph of a Commutative Ring Without Identity”. International Electronic Journal of Algebra 23/23 (January 2018), 176-202. https://doi.org/10.24330/ieja.373663.
JAMA Anderson DF, Weber D. The zero-divisor graph of a commutative ring without identity. IEJA. 2018;23:176–202.
MLA Anderson, David F. and Darrin Weber. “The Zero-Divisor Graph of a Commutative Ring Without Identity”. International Electronic Journal of Algebra, vol. 23, no. 23, 2018, pp. 176-02, doi:10.24330/ieja.373663.
Vancouver Anderson DF, Weber D. The zero-divisor graph of a commutative ring without identity. IEJA. 2018;23(23):176-202.