×

Some properties of the complement of the zero-divisor graph of a commutative ring. (English) Zbl 1230.13012

Let \(R\) be a commutative ring with identity. Let \(\Gamma(R)\) be the zero-divisor graph of \(R\), whose vertices are the nonzero zero-divisors, with two distinct vertices \(x\) and \(y\) being adjacent if and only if \(xy=0\). Let \(G=\Gamma(R)^c\) be the complement graph of \(\Gamma(R)\). The author proves that if \(G\) is connected, then its girth is three and its radius is two. The author also studies the cliques in \(G\). In particular, he characterizes when the clique number of \(G\) is finite under various assumptions on the ring \(R\).

MSC:

13A99 General commutative ring theory
05C75 Structural characterization of families of graphs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 217, no. 2, pp. 434-447, 1999. · Zbl 0941.05062 · doi:10.1006/jabr.1998.7840
[2] D. D. Anderson and M. Naseer, “Beck’s coloring of a commutative ring,” Journal of Algebra, vol. 159, no. 2, pp. 500-514, 1993. · Zbl 0798.05067 · doi:10.1006/jabr.1993.1171
[3] D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, “The zero-divisor graph of a commutative ring. II,” in Ideal theoretic methods in commutative algebra, vol. 220 of Lecture Notes in Pure and Applied Mathematics, pp. 61-72, Dekker, New York, NY, USA, 2001. · Zbl 1035.13004
[4] D. F. Anderson, R. Levy, and J. Shapiro, “Zero-divisor graphs, von Neumann regular rings, and Boolean algebras,” Journal of Pure and Applied Algebra, vol. 180, no. 3, pp. 221-241, 2003. · Zbl 1076.13001 · doi:10.1016/S0022-4049(02)00250-5
[5] D. F. Anderson and S. B. Mulay, “On the diameter and girth of a zero-divisor graph,” Journal of Pure and Applied Algebra, vol. 210, no. 2, pp. 543-550, 2007. · Zbl 1119.13005 · doi:10.1016/j.jpaa.2006.10.007
[6] D. F. Anderson and A. Badawi, “On the zero-divisor graph of a ring,” Communications in Algebra, vol. 36, no. 8, pp. 3073-3092, 2008. · Zbl 1152.13001 · doi:10.1080/00927870802110888
[7] M. Axtell, J. Coykendall, and J. Stickles, “Zero-divisor graphs of polynomials and power series over commutative rings,” Communications in Algebra, vol. 33, no. 6, pp. 2043-2050, 2005. · Zbl 1088.13006 · doi:10.1081/AGB-200063357
[8] F. Azarpanah and M. Motamedi, “Zero-divisor graph of C(X),” Acta Mathematica Hungarica, vol. 108, no. 1-2, pp. 25-36, 2005. · Zbl 1092.54007 · doi:10.1007/s10474-005-0205-z
[9] I. Beck, “Coloring of commutative rings,” Journal of Algebra, vol. 116, no. 1, pp. 208-226, 1988. · Zbl 0654.13001 · doi:10.1016/0021-8693(88)90202-5
[10] R. Levy and J. Shapiro, “The zero-divisor graph of von Neumann regular rings,” Communications in Algebra, vol. 30, no. 2, pp. 745-750, 2002. · Zbl 1055.13007 · doi:10.1081/AGB-120013178
[11] T. G. Lucas, “The diameter of a zero divisor graph,” Journal of Algebra, vol. 301, no. 1, pp. 174-193, 2006. · Zbl 1109.13006 · doi:10.1016/j.jalgebra.2006.01.019
[12] S. B. Mulay, “Cycles and symmetries of zero-divisors,” Communications in Algebra, vol. 30, no. 7, pp. 3533-3558, 2002. · Zbl 1087.13500 · doi:10.1081/AGB-120004502
[13] S. P. Redmond, “Central sets and radii of the zero-divisor graphs of commutative rings,” Communications in Algebra, vol. 34, no. 7, pp. 2389-2401, 2006. · Zbl 1105.13007 · doi:10.1080/00927870600649103
[14] K. Samei, “The zero-divisor graph of a reduced ring,” Journal of Pure and Applied Algebra, vol. 209, no. 3, pp. 813-821, 2007. · Zbl 1108.13009 · doi:10.1016/j.jpaa.2006.08.008
[15] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New York, NY, USA, 2000. · Zbl 0938.05001
[16] W. Heinzer and J. Ohm, “On the Noetherian-like rings of E. G. Evans,” Proceedings of the American Mathematical Society, vol. 34, pp. 73-74, 1972. · Zbl 0239.13014 · doi:10.1090/S0002-9939-1972-0294316-2
[17] W. Heinzer and J. Ohm, “Locally noetherian commutative rings,” Transactions of the American Mathematical Society, vol. 158, pp. 273-284, 1971. · Zbl 0223.13017 · doi:10.2307/1995903
[18] S. Visweswaran, “Some results on the complement of the zero-divisor graph of a commutative ring,” Journal of Algebra and its Applications, vol. 10, no. 3, pp. 573-595, 2011. · Zbl 1228.13012 · doi:10.1142/S0219498811004781
[19] T. W. Hungerford, Algebra, vol. 73 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1980. · Zbl 0442.00002
[20] N. Ganesan, “Properties of rings with a finite number of zero divisors,” Mathematische Annalen, vol. 157, pp. 215-218, 1964. · Zbl 0135.07704 · doi:10.1007/BF01362435
[21] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, Reading, Mass, USA, 1969. · Zbl 0175.03601
[22] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, Ill, USA, 1974. · Zbl 0203.34601
[23] R. Gilmer, Multiplicative ideal theory, Dekker, New York, NY, USA, 1972. · Zbl 0248.13001
[24] S. Visweswaran, “ACCR pairs,” Journal of Pure and Applied Algebra, vol. 81, no. 3, pp. 313-334, 1992. · Zbl 0768.13003 · doi:10.1016/0022-4049(92)90063-L
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.