GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

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,

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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.

International Electronic Journal of Algebra-Cover

ISSN: 1306-6048
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2007
Yayıncı: Abdullah HARMANCI

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