GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

Year 2020, Volume: 27 Issue: 27, 237 - 262, 07.01.2020

David F. Anderson Grace Mcclurkin

https://doi.org/10.24330/ieja.663079

Cited By: 5

Abstract

Let $R$ be a commutative ring with $1 \neq 0$ and $Z(R)$ its
set of zero-divisors. The zero-divisor graph of $R$ is the (simple) graph $\Gamma(R)$ with vertices $Z(R) \setminus \{0\}$, and distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. In this paper, we consider generalizations of $\Gamma(R)$ by modifying the vertices or adjacency relations of $\Gamma(R)$. In particular, we study the extended zero-divisor graph $\overline{\Gamma}(R)$, the annihilator graph $AG(R)$, and their analogs for ideal-based and congruence-based graphs.

Keywords

Zero-divisor graph, commutative ring with identity

References

• M. Afkhami, N. Hoseini and K. Khashyarmanesh, The annihilator ideal graph of a commutative ring, Note. Mat., 36(1) (2016), 1-10.
• M. Afkhami, K. Khashyarmanesh and Z. Rajabi, Some results on the annihilator graph of a commutative ring, Czechoslovak Math. J., 67(1) (2017), 151-169.
• M. Afkhami, K. Khashyarmanesh and S. M. Sakhdari, The annihilator graph of a commutative semigroup, J. Algebra Appl., 14(2) (2015), 1550015 (14 pp).
• D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
• 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, S.-E. Kabbaj, B. Olberding, I. Swanson, 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, Groups, modules, and model theory - surveys and recent developments, Springer, 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, Abian's poset and the ordered monoid of annihilator classes in a reduced commutative ring, J. Algebra Appl., 13(8) (2014), 1450070 (18 pp).
• D. F. Anderson and J. D. LaGrange, Some remarks on the compressed zero- divisor graph, J. Algebra, 447 (2016), 297-321.
• 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 E. F. Lewis, A general theory of zero-divisor graphs over a commutative ring, Int. Electron. J. Algebra, 20 (2016), 111-135.
• 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 and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210(2) (2007), 543-550.
• S. E. Atani, A. Y. Darani and E. R. Puczylowski, On the diameter and girth of ideal based zero-divisor graphs, Publ. Math. Debrecen, 78(3-4) (2011), 607-612.
• A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(3) (2007), 417-429.
• A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42(1) (2014), 108-121.
• I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
• D. Bennis, J. Mikram and F. Taraza, On the extended zero divisor graph of commutative rings, Turkish J. Math., 40(2) (2016), 376-388.
• B. Bollobas, Graph Theory, An Introductory Course, Graduate Texts in Math- ematics, 63, Springer-Verlag, New York-Berlin, 1979.
• J. Coykendall, S. Sather-Wagstaff, L. Sheppardson and S. Spiroff, 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. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum, 65(2) (2002), 206-214.
• F. DeMeyer and K. Schneider, Automorphisms and zero-divisor graphs of com- mutative rings, Internat. J. Commutative Rings, 1(3) (2002), 93-106.
• S. Dutta and C. Lanong, On annihilator graphs of a finite commutative ring, Trans. Comb., 6(1) (2017), 1-11.
• R. Gilmer, Commutative Semigroup Rings, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, 1984.
• J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Text- books in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
• I. Kaplansky, Commutative Rings, Revised edition, The University of Chicago Press, Chicago, 1974.
• G. McClurken, Generalizations and Variations of the Zero-Divisor Graph, Doctoral dissertation, The University of Tennessee, August 2017.
• S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30(7) (2002), 3533-3558.
• Sh. Payrovi and S. Babaei, The compressed annihilator graph of a commutative ring, Indian J. Pure Appl. Math., 49(1) (2018), 177-186.
• S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra, 31(9) (2003), 4425-4443.
• J. G. Smith, Jr., When ideal-based zero-divisor graphs are complemented or uniquely complemented, Int. Electron. J. Algebra, 21 (2017), 198-204.
• S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39(7) (2011), 2338-2348.