Yıl 2018, Cilt 23, Sayı 23, Sayfalar 176 - 202 2018-01-11

The zero-divisor graph of a commutative ring without identity

David F. Anderson [1] , Darrin Weber [2]

64 82

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.
Zero-divisor graph, commutative ring without identity
  • 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.
Konular
Dergi Bölümü Makaleler
Yazarlar

Yazar: David F. Anderson

Yazar: Darrin Weber

Bibtex @araştırma makalesi { ieja373663, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2018}, volume = {23}, pages = {176 - 202}, doi = {10.24330/ieja.373663}, title = {The zero-divisor graph of a commutative ring without identity}, key = {cite}, author = {Anderson, David F. and Weber, Darrin} }
APA Anderson, D , Weber, D . (2018). The zero-divisor graph of a commutative ring without identity. International Electronic Journal of Algebra, 23 (23), 176-202. DOI: 10.24330/ieja.373663
MLA Anderson, D , Weber, D . "The zero-divisor graph of a commutative ring without identity". International Electronic Journal of Algebra 23 (2018): 176-202 <http://dergipark.gov.tr/ieja/issue/33727/373663>
Chicago Anderson, D , Weber, D . "The zero-divisor graph of a commutative ring without identity". International Electronic Journal of Algebra 23 (2018): 176-202
RIS TY - JOUR T1 - The zero-divisor graph of a commutative ring without identity AU - David F. Anderson , Darrin Weber Y1 - 2018 PY - 2018 N1 - doi: 10.24330/ieja.373663 DO - 10.24330/ieja.373663 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 176 EP - 202 VL - 23 IS - 23 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.373663 UR - http://dx.doi.org/10.24330/ieja.373663 Y2 - 2018 ER -
EndNote %0 International Electronic Journal of Algebra The zero-divisor graph of a commutative ring without identity %A David F. Anderson , Darrin Weber %T The zero-divisor graph of a commutative ring without identity %D 2018 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 23 %N 23 %R doi: 10.24330/ieja.373663 %U 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 (Ocak 2018): 176-202. http://dx.doi.org/10.24330/ieja.373663