The zero-divisor graph of a commutative ring without identity

Year 2018, , 176 - 202, 11.01.2018

David F. Anderson , Darrin Weber

https://doi.org/10.24330/ieja.373663

Cited By: 10

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.

Keywords

Zero-divisor graph, commutative ring without identity

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.